Set: Explanation

Set is synonymous with the words, ‘collection’, aggregate’, ‘class’, and is comprised of elements.

The words ‘element’, ‘object’, and ‘member’ are synonymous.

Sets designated by specific letters.

N: natural numbers
Z : integers
Z⁺: positive integers
Q: rational numbers
Q⁺: positive rational numbers
R: real numbers
R⁺: positive real numbers
C: complex numbers

Description of a set

Sets can be described by roster method or set-builder method.

Roster method:
In this method, the set is described by listing all the elements within braces { }, separated by commas.

Example: {2,4,6,8,10}
It is a set having 5 elements.

Set-builder method:
In this method, a set is described by a property of x where x represents the elements. If the property of x is represented by P(x), the set description is given by
{x : P(x) is satisfied} or {x| P(x) is satisfied}

Example: {x| x is an even number less than or equal to 10}
This description will give {2,4,6,8,10} in roster form.

Types of sets

Empty set (ф)
A set is said t be empty or null or void set if it has no element and it is denoted by ф.

Singleton set
A set consisting of single element.

Finite set
A set is called a finite set if it is either void set or its elements can be listed (counted or labeled) by natural numbers 1,2,3 … and the counting of number of elements stops at a certain natural number of say (n).

The number of elements in a finite set (n) is called the cardinal number or order of a finite set A and is denoted by n(A).

Infinite set
A set who elements cannot be listed by the natural numbers however large the number may be is called an infinite set.

Equivalent set
Two finite sets are equivalent if their cardinal numbers or number of elements are same.

Equal set
Two sets A and B are equal if every element in A is a member of B and every element of B is a member of A.

Subset
When A and B are two sets, if every element of A is an element of B, then A is called a subset of B.

Universal set (U)
In discussions of sets, the superset that contains all other sets in discussion is called the universal set.

Power set
When A is a set, the collection or family of all subsets of A is called the power set of A and is denoted by P(A).

Power set is a set of subsets or elements of a power set are subsets of a set.
P(A) = {S: S is a subset of A}

If A is a finite set having n elements, the P(A) has 2ⁿ elements.


Complement of a set
If U is a universal set, the complement of a set A with respect to U is denoted as A’ or A^c or U – A . It is a set of those elements of U which are not in A.

A’ = {x| x є U, and x is does not belong to A}

Theorems on subsets

1. Every set is a subset of itself.
2. The empty set is a subset of every set.
3. The total number of subsets of a finite set containing n elements is 2ⁿ

Operations on sets

Union of sets
The union of sets A and B is th set of all those elements which belong either to A or to B or to both A and B.

The symbol used to denote union of sets and A and B is A U B.

x Є (A U B) implies x Є A or x Є B
x does not belong to A U B implies x does not belong to A and also x does not belong to B.

Intersection of sets
The intersection of sets A and B is the set of all those elements that belong to both A and B.

The intersection of sets A and B is denoted by A ∩ B.

x Є (A ∩ B) implies x Є A and also x Є B.


Difference of sets
The difference of sets A and B, written as A-B is the set of all those elements of A which do not belong B.

It means x Є (A - B) implies x Є A or x does not belong to B.

Symmetric difference of sets
The symmetric difference of sets A and B is the set (A-B)U((B-A) and is denoted by AΔB.

AΔB = (A-B)U((B-A) = {x: x does not belong to A ∩ B).

Types of sets based on operations

Disjoint sets

Complement of a set
When U is the universal set and A is a subset of U, the complement of A with respect to U is denoted by A’ or A₀ or U-A and it is defined as the set of all those elements of U which are not in A.

A’ = {x: x does not belong to A but x ЄU}
x ЄA’ implies x does not belong to A.

Laws of algebra of sets

1. Idempotent laws

(i) A U A = A
(ii) A ∩ A = A

2. Identity laws

(i) A U ф = A
(ii) A ∩ U = A

3. Commutative law
(i) A U B = B U A
(ii) A ∩ B = B ∩ A

4. Associative laws

(i) (A U B) U C = A U (B U C)
(ii) (A ∩ B) ∩ C = A ∩ (B ∩ C)

5. Distributive laws

(i) A U (B ∩ C) = (A U B) ∩ (A U C)
(ii) A ∩ (B U C) = (A ∩ B) U (A ∩ C)

6. De-morgan’s laws

(i) (A U B)’ = A’ ∩ B’
(ii) (A ∩ B)’ = A’ U B’

Some more deductions/theorems/ related to operations on sets

If A and B are two sets

(i) A – B = A ∩ B’
(ii) B – A = B ∩ A’
(iii) A – B = A <=> A ∩ B = ф
(iv) (A – B) U B = A U B
(v) (A-B) ∩ B = ф
(vi) A is a sub set of B <=> B’ is a subset of A’
(vii) (A-B) U (B-A) = (A U B) – (A ∩ B)

If A, B and C are three sets, then

(i) A – (B ∩ C) = (A-B) U (A-C)
(ii) A – (B U C) = (A-B) ∩ (A-C)
(iii) A ∩ (B-C) = (A ∩ B) - (A ∩ C)
(iv) A ∩ (B Δ C) = (A∩B) Δ (A∩C)

Number of elements in sets n(A) and Some Results on Them

Note union operation and universal set have the same symbol in these pages. Hence identify appropriately.

n(A) denotes the number of elements in the set A. Similarly n(B) and n(C).

If A,B and C are finite sets. U is the finite universal set, then

(i) n(A U B) = n(A) +n(B) – n(A∩B)

(ii) n(A U B) = n(A) +n(B) <=> A, B are disjoint non-void sets.

(iii) n(A-B) = n(A) –n(A∩B)

(iv) n(A ΔB) = Number of elements which belong to exactly one of A or B
= n((A-B) U (B-A))

(v) n(A U B U C) = n(A) + n(B) + n(C) – n(A∩B) – n(B∩C) – n(A∩C)+n(A∩B∩C)

(vi) No. Of elements in exactly two of the sets A,B,C
= n(A∩B) + n(B∩C)+n(C∩A)-3n(A∩B∩C)

(vii) No. Of elements in exactly one of the sets A,B,C
= n(A) +n(B)+n(C)-2n(A∩B)-2n(B∩C)-2n(A∩C)+3n(A∩B∩C)

(viii) n(A’ U B’) = n((A∩B)’) = n(U) – n(A∩B)

(ix) n(A’∩B’) = n((AUB)’) = n(U)-n(A∩B)

Ordered Pair

An Ordered pair consists of two objects or elements in a given fixed order.

For example when A and B are any two sets, a pair (a,b) where a ЄA and b ЄB is an ordered pair. The fixed order comes from the two sets A and B and the first element is from A and the second element is from B.

Equality of ordered pairs: Two ordered pairs (a1,b1) and (a2,b2) are equal to if a1 = a2 and b1 = b2.

Cartesian product of sets

Cartesian product is an operation on sets.

Let A and B be any two non empty sets. The set of all ordered pairs (a,b) such that aЄA and bЄB is called the Cartesian product of the sets A and B and is denoted by A×B

A×B represents Cartesian product of sets and it is a set of ordered pairs.

Theorems on Cartesian product of sets

Theorem 1; For any three sets

(i) A×(B U C) = (A×B) U (A×C)
(ii) A×(B∩C) = (A×B) ∩(A×C)

Theorem 2: For any three sets

A×(B – C) = (A×B) – (A×C)

Theorem 3: If and A and B are any two non-empty sets, then

A×B = B×A  A = B

Theorem 4: If A is a subset of B, A×A is a sub set of (A×B) ∩(B×A)

Theorem 5
If A is a subset of B, (A×C) is a subset of (B×C) for any set C.

Theorem 6
If A is a subset of B and C is a subset of D, (A×C) is a subset of (B×D)

Theorem 7
For any sets A,B,C , D,

(A×B) ∩(C×D) = (A∩C) ×(B∩D)

Theorem 8

For any three sets A,B,C

i. (A×(B’ U C’) = (A×B) ∩(A×C)
ii. (A×(B’ ∩ C’) = (A×B) U(A×C)

Theorem 9
When A and B are two non-empty sets having n elements in common, (A×B) and (B×A) will have n² elements in common.

Relation - Definition

Let A and B be two sets. Then a relation R from A to B is a subset of A×B.

R is a relation from A to B <=> R is a subset of A×B.

Total number of relations: If A and B are two non empty sets with m and n elements respectively, A×B consists of mn ordered pairs.
Since each subset defines a relation from A to B, so total number of relations from A to B is 2^mn.

Domain and Range of Relation

R is a relation means that it is a set of ordered paris.


Domain of a relation

When R is a relation from a set A to set B, the set of all first components of the ordered pairs belonging to R is called the domain of R.

Range of a relation

When R is a relation from a set A to set B, the set of all second components of the ordered pairs belonging to R is called the range of R.

In a relation from set A to set B, domain of the relation will be a subset of A and range of the relation will be a subset of B.

Relation on Set A and Inverse Relation

Relation on Set A
A relation from A to A i.e., a subset of A×A, is called a relation on set A.

Inverse relation
When a relation R is from set A to set B, a relation from set B to set A denoted by R^-1 is the inverse of R. That is if R is a set of (a,b), then R^-1 is a set of (b,a).

In the case Domain of relation R = Range of relation R^-1

Range of relation R = Domain of relation R^-1

Types of relations

Void or empty relation
When A is a set, ф is a subset of A×A and so it is a relation on A. This relation is called the void or empty relation on A.

Universal relation
When A is a set, A×A is a subset of A×A and so it is a relation on A. This relation is called the universal relation.

Identity relation
When A is a set, the relation I_A = {(a,a):a ЄA} on A is called the identity relation on A.

Reflexive relation
A relation on a set A is said to be reflexive if every element of A is related to itself.

Think. What is the difference between identity relation and reflexive relation?

Symmetric relation
A relation R on a set A is said to be a symmetric relation iff

(a,b) Є R implies (b,a) ЄR for all a,b Є A.

i.e. aRb implies bRa for a,b Є.

Transitive relation
A relation R on a set A is said to be a transitive relation iff
(a,b) ЄR and (b,c) ЄR implies (a,c) ЄR for a,b,c ЄA.

Antisymmetric relation
A relation R on set A is said t be an antisymmetric relation iff
(a,b) Є R and(b,a) Є R implies a =b for all a,b ЄA.

Equivalence relation
A relation R on set A is said to be an equivalence relation on A iff
i. it is reflexive.
ii. it is symmetric
iii. it is transitive.

Some more properties and results on relations

1. If R and S are two equivalence relations on a set A, then R∩S is also an equivalence relation on A.
2. The union of two equivalence relations on a set is not necessarily an equivalence relation on the set.
3. If R is an equivalence relation on a set A, the R^-1 is also an equivalence relation on A.

Composition of Relations

When r and S are two relations from set A to B and B to C respectively, we can define a relation SoR from A to C such that

(a.c) Є SoR imples for all b Є B subject to the relations (a,b) ЄR and (b.c) ЄS.

SoR is called the composition of R and S.

Properties of SoR

In general RoS is not equal to SoR.

(SoR) ^- = R^-oS^-

Function – definition

Let A and B be two non-empty sets. Then a function ‘f’ from set A to set B is a rule or method or correspondence which associates elements of set A to elements of set B such that

(i) all elements of set A are associated to elements in set B.
(ii) an element of set A is associated to a unique element in set B.

Domain, Co-Domain and Range of a Function

Let f: A → B. Then, the set A is known as the domain of f and the set B is known as the co-domain of f.

The set of f-images of elements of A (elements in set B associated with elements in set A) is known as the range of f or image set of A under f and is denoted by f(A)

Thus range of f = f(A) = {f(x): x Є A}

F(A) is a subset of B.

Range of f is a subset of co-domain of f.

Description of a Function

A function f:A→B can be described by giving A and f(a) for every element in A if it is a finite set.

If A is infinite, the functions are described by a formula.
For example A function f:Z→Z is given by the formula f(x) = x²+1

Equal functions

Two functions f and g are equal or f = g iff
The domain of f = domain of g,
The co-domain of f = the codomain of g and
f(x) = g(x) for every x belonging to their common domain.

Number of functions

Let A and B be two finite sets having m and n elements.
The total number of functions from A to B is n^m.

Function as a relation

A relation f from A to B (both being non empty sets), that is a sub set of A×B is called a function from A to B if
For each a ЄA there exists b ЄB such that (a,b) Єf and if (a,b) Єf and (a,c) Єf means b = c.

Kinds of functions

Kinds of Functions

One-One function

A function f:A→B is said to be a one-one function if different elements of A have different images in B.

Many-One function

A function f:A→B is said to be many-one function if two or more elements of set A have the same image in B.

Onto function

A function f:A→B is said to be an onto function if every element of B is the f image of some element of A.

Range of is the codomain of f. Codomain of f is B.

Into function

A function f:A→B is said to be an into function if there exists an element in B having no pre-image in A.

One-one onto function

A function which is one-one as well as onto function.

Composition of functions

If f:A→B and g:B→C are two functions, then a function gof: A→C is defined by

(gog)(x) = g(f(x), for all x ЄA and is called the composition of f and g.

Properties of composition of functions

The composition of functions is not commutative fog ≠ gof.

The composition of functions is associative. If f,g, and h are functions such that (fog)oh and fo(goh) exist, then
(fog)oh = fo(goh)

The composition of two bijections is also a bijection.

Inverse of an element

In case of f:A→B, if a ЄA is associated with b ЄB, then b is called the f image of ‘a’ and is written as b = f(a). a is said to be the pre-image or inverse element of ‘b’ under f and we write a = f^-1(b).

Inverse of a function

If If f:A→B is a bijection meaning x ЄA and y ЄB then we associate y ЄB with x ЄA and such a function is known as the inverse function and is denoted by f^-1.

Properties of inverse of a function

The inverse of a bijection is unique

The inverse of a bijection is also a bijection

Binary Operations and Types of Binary Operations

When S is a nonempty set, a function f:S×S→S is called a binary operation.

Each ordered pair (a,b)Є(S×S) is associated to a unique element f(a,b) in S.

Examples: addition of a and b (both natural numbers). a+b is also a natural number.

a belongs to N, b belongs to N, and a+B also belongs to N.


Types


Commutative binary operation

Associative binary operation

Distributive binary operation

Logarithm - Definition

If a^x = y then log_ay = x

log is the abbreviation of the word logarithm

Fundamental Laws of Logarithms

log MN = log M + log N

log M/N = log M - log N

log Mⁿ = n log M

log 1 = 0

log_aa = 1 (log of any positive quantity of the same base is always one.

Systems of Logarithms

Common logarithms to the base 10

Natural logarithms to the base e

Standard Form of Decimal

If k is a positive number we can express it as

k = m*10^p

Where m is a decimal number such that 1≤m≤10 and p is an integer.

This form m*10^p is called standard form of decimal.

Finding Log N from tables

Find the characteristic of N.
Find the mantisssa
Log N = Charateristic + Mantissa

Antilogarithms

If log n = m

n = antilog m

Definition of sets of numbers

N = set of natural numbers

I = set of integers

Q = set of all rational numbers

R = set of real numbers rational and irrational

C = set of complex numbers

Imaginary unity - Iota

Sqrt(-1) = i
“i” is called imaginary unity

Integral powers of IOTA (i)

i³ = i*i² = i*(-1) = -i

Imaginary quantities

Square roots of numbers -3, -5 etc are called imaginary quantities

Complex numbers - definition

Number of the form a+ib (ex: 4+i3) is called a complex number.

a is called real part Re(z) and b is called imaginary part Im(z).

Equality of complex numbers

Two complex numbers z1 = a1+ib1 and z2 = a2+ib2 are equal if a1 = a2 and b1 = b2.

It means z1 = z2 if Re(z1) = Re(z2) and Im(z1) = Im(z2).

Addition of complex numbers

When z1 = a1+ib1 and z2 = a2+ib2 the addition of the two z1 + z2 is defined as the complex number (a1+a2) + i(b1+b2)

Re(z1+z2) = Re(z1) + Re(z2)

Im(z1+z2) = Im(z1)+Im(z2)

Subtraction of complex numbers

When z1 = a1+ib1 and z2 = a2+ib2 the subtraction of the two z1 - z2 is defined as z1+(-z2)

Multiplication of complex numbers

(a1+ib1) (a2+ib2) by multiplying and simplifying we get

(a1a2 – b1b2) + i(a1b2+a2b1)

Multiplicative inverse of a+ib = a/(a² + b²) - ib/(a² + b²))

Division of complex numbers

z1/z2 = z1* Multiplicative inverse of z2

Conjugate of a complex number

conjugate of z (= a+ib) = a-ib (is termed as z bar)

Modulus of a complex number

|z| = |a+ib| = SQRT(a² +b²)

Properties of Modulus

If z is a complex number, then

(i) |z| = 0 <=> z = 0
(ii) |z| = |conjugate of z| = |-z| = |-conjugate of z|
(iii) -|z| ≤Re(z) ≤|z|
(iv) -|z| ≤Im(z) ≤|z|
(v) z*congulage of z = |z|²

Reciprocal of a complex number

Multiplicative inverse and reciprocal are same

Complex number as a rotating arrow in the argand plane

Roots of a complex number

Write the give number in polar form

Add 2mπ to the argument

Apply the De Moivre's theorem

Put m = 0,1,2,...,(n-1) i.e. one less than the number in the denominator of the given index in the lowest form.

Sequences and Series - Definitions

Sequence

A sequence is a function whose domain is the set N of natural numbers.

Sequence is denoted by ‘a’ and the nth term in the sequence a(n) is denoted by an/sub>.

A sequence whose range is a subset of R is called a real sequence.

Representation of a sequence

One way is to list its first few terms till the rule for writing down other terms becomes clear.

Another way is to represent a real sequence is to give a rule of writing the nth term of the sequence.

Series

If a1, a2, a3, … is a sequence, then the expression a1+a2+a3+… is a series.


Progressions

It is not necessary that the terms of a sequence always follow a certain pattern or they are described by some explicit formula for the nth term. Those sequences whose terms follow certain patterns are called progressions.

General term of an A.P.

nth term = an = a+(n-1)d

Selection of terms in an A.P.

a-d, a, a+d

Sum of n terms of an A.P.

Sn = n(a+1)/2

Sum of n terms of a G.P.

S_n = a[(rⁿ-1)/(r-1)]

Sum of an Infinite G.P

S = a/(1-r)
r is less than one

Sum to n terms - some special sequences

1. Natural numbers

n(n+1)/2

2. Squares of natural numbers

n(n+1)(2n+1)/6

3. Cubes of natural numbers

[n(n+1)/2]²

Harmonic progression

A sequence a1, a2, a2,…,an of non-zero numbers is called a Harmonic progression if the sequence 1/a1,1,a2,..,1/an,.. is an A.P.

Example: The sequence 1,1/4,1/7,1/10,… is a H.P. because the sequence 1,4,7,10,… is in A.P.

d of the corresponding AP = 1/a2 -1/a1

an of H.P. 1/[a+(n-1)d] where a = 1/a1

Insertion of n harmonic means between two give numbers a and b

a,H1,H2,…,Hn,b are in H.P.
=> 1/a, 1/H1,1/H2,…,1/Hn,1/b are in A.P.

Let d be common difference of this A.P.
The last term in AP 1/b is the (n+2)th term.

So 1/b = 1/a +(n+1)d
=> d = (1/b -1/a)/(n+1) = (a-b)/ab(n+1)
=> 1/H1 = (1/a) +d
1/H2 = (1/a)+2d

1/Hn = (1/a)+nd

Harmonic mean of n numers

If a1, a2, ..., an are n non-zero numbers, then the harmonic mean H of these numbers is given by

1/H = [1/a1 +1/a2+…+1/an]/n

Properties of arithmetic, geometric, and harmonic means between two given numbers (a and b)

A.M. = A = (a+b)/2
G.M. = G = √ab
H.M. = H = 2ab/(a+b)

1. A≥G≥H
2. A,G,H form a GP, i.e., G² = AH
3. The equation x²-2Ax+G² has as its roots a and b.
4. The equation x³-3A x²+3G³x/H-G³ = 0 has as its roots a,b,c when A,G,and H are arithmetic mean, geometric mean, and harmonic mean of three numbers a,b,and c.

Quadratic Equations and Expressions - Definitions

Real Polynomials: Coefficients are real numbers and variables take real values.

Complex Polynomials: Coefficients are complex numbers and variable is varying complex number.

Polynomial equation:
f(x) = 0

Roots of an equation: The values of the variable satisfying the given equation are called its roots.

Some conclusions on Roots of a Polynomial Equation

1. An equation of degree n has n roots, real or imaginary.
2. Surd and imaginary roots always occur in pairs, i.e. if 5-3i is a root of an equation, then 5 +3i is also its root. Similarly, if 3+SQRT(5) is a root of a given equation, then 3-SQRT(5) is also its root.
3. An odd degree equation has at least one real root, whose sign is opposite to that of its last term, provided that the coefficient of highest degree term is positive.
4. Every equation of an even degree whose constant term is negative and the coefficient of highest degree term is positive, has at least two real roots, one positive and one negative

Descartes Rule of Signs

The maximum number of positive real roots of a polynomial equation f(x) = 0 is the number of changes of signs from positive to negative and negative to positive in f(x).

Relations between roots and coefficients

S1 = α1+α2+...+αn = -a1/a0

S2 = α1α2 + α1α3+... = Σαiαj; i≠j = (-1)² (a2/a0)


Sn = α1α2...αn = (-1)ⁿ(const. term/a0) {constant term = an)

Formation of a polynomial equation from given roots

If α1, α2, α3,...,αk are the roots of an nth degree equation

xⁿ-S1x^n-1+S2x^n-2-S3x^n-3++...+(-1)ⁿSn = 0

where Sk denotes the sum of the products of roots taken k at a time.

Roots of a quadratic equation with real coefficients

ax²+bx+c where a≠0, a,b,c Є R is a quadratic equation with real coefficients.

The quantity D = b²-4ac is the called the discriminant of the quadratic equation.

1. The roots are real and distinct if and only if D>0.
2. The roots are real and equal if and only D = 0
3. The roots are complex with non-zero imaginary part if and only if D<0.
4. The roots are rational iff a,b,c are rational and D is a proper square.
5. The roots are of the form p+√q (p,q Є Q), iff a,b,c are rational and D is not a perfrect square.
6. If a =1, b,c ЄI and the roots are rational numbers, then these roots must be integers.
7. If a quadratic equation in x has more than two roots, then it is an identity in x that is a=b=c=o.

Graph of Quadratic Expression

Graph of a quadratic expression is a parabola.

Sign of a quadratic expression for real values of the variable

For real values of x, the sign of the quadratic expression f(x) = ax² +bx+c is the same as that of 'a' except when the roots of the equation ax²+bx+c are real and distinct and x lies between them.


ax²+bx+c is greater than 0 for all x Є R iff a is greater than 0 and D is less than zero. (D is discriminant b²-ac), and

ax²+bx+c is lesser than 0 for all x Є R iff a is less than 0 and D is less than zero.

Values of a rational expression P(x)/Q(x) for real values of x, where P(x) and Q(x) are quadratic expressions

Factorial

Factorial: the continued product of first n natural numbers is called the “n factorial” and is denoted by n! or

i.e. n! = 1*2*3…*(n-1)*n

4! = 1*2*2*4

n! is defined for positive integers only.

0! is defined as 1.

n! = n*(n-1)!

Exponent of prime number p in factorial of n (n!)

The exponent of prime number of 3 in 100! is 48.
It means 100! is divisible by 3⁴⁸

How do you find it?
Let p be a prime number and n be a positive integer. Then find the last integer in the sequence 1,2,…,n which is divisible by p.
Express this integer as [n/p]p.
[n/p] denotes the greatest integer less than or equal to n/p

In case of 3 (p) and 100 (n); [n/p] is 33 and n/p is 33 and 1/3.

Let E_p(n) denote the exponent of the prime p in the positive integer n. Then,


E_p(n!) = E_p(1.2.3…(n-1).n)

This will be equal to E_p(p.2p.3p…[n/p]p)
= [n/p]+ E_p(1.2.3...[n/p])

This process continues and we get the answer

E_p(n!) = [n/p] + [n/p²]+…+[n/p^s]
Where s is the largest positive integer such that p^s≤n≤p^s+1

Hence applying the formula to find exponent of prime 3 in 100!

E₃(100!) = [100/3] + [100/3²] + [100/3³] + [100/3⁴]
= 33+11+3+1 = 48

Note: remember the meaning of notation [100/3] or [n/p]

Fundamental principle of multiplication

If there are two jobs such that one of them can be completed in m ways, and when it has been completed in any one of these m ways, second job can be completed in n ways, then the two jobs in succession can be completed in m*n ways.

Fundamental principle of addition

If there are two jobs such that they can be performed independently in m and n ways respectively, then either of the two jobs can be performed in (m+n) ways.

Permutations Definition and Theorems

Each of the arrangement which can be made by taking some or all of a number of things is called a permutation.



Theorem 1

Let r and n be positive integers such that 1≤r≤n. then the number of all permutations of n distinct things taken r at a time is given by

n(n-1)(n-2)…(n-(r-1))

Notation: Let r and n be positive integers such that 1≤r≤n. then the number of all permutations of n distinct things taken r at a time is denoted by the symbol P(n,r) or ⁿ C_r.

Then P(n,r) = ⁿ C_r = n(n-1)(n-2)…(n-(r-1))


Theorem 2

P(n,r) = ⁿ C_r = n!/(n-r)!

Theorem 3

The number of all permutations of n distinct things taken all at a time is n!.

Theorem 4

0! = 1

Permutations under certain conditions

Three theorems

Theorem 1
The number of all permutations of n different objects taken r at a time, when a particular object is to be always included in each arrangement is r.^n-1C_r-1

Theorem 2

The number of all permutations of n different objects taken r at a time, when a particular object is never taken in each arrangement is, ^n-1C_r-1

Theorem 3

The number of all permutations of n different objects taken r at a time, when two specified objects always occur together is 2!(r-1) ^n-2C_r-2

Permutations of Objects not all Distinct

Theorems and Formulas

Theorem
The number of mutually distinguishable permutations of n things, taken all at a time, of which p are alike of one kind, q alike of second such that p+q = n, is

n!/p!q!


Formulas based on the above theorem

1. The number of mutually distinguishable permutations of n things, taken all at a time, of which p1 are alike of one kind, p2 alike of second,…, pr alike of of rth kind such that p1+p2+…pr = n, is

n!/p1!p2!…pr!

2. The number of mutually distinguishable permutations of n tings, of which p are alike of one kind, q alike of second and remaining all are distinct is
n!/p!q!

3. suppose there are r things to be arranged, allowing repetitions. Let further p1,p2,…,pr be the integers such that the first object occurs exactly p1 times, the second occurs exactly p2 times, etc. Then the total number of permutations of these r objects to the above condition is

(p1+p2+…+pr)!/p1!p2!…pr!

Permutations when Objects can Repeat

Theorem
The number of permutations of n different things, taken r at a time, when each may be repeated any number of times in each arrangement is n² .

Circular Permutations

If we arrange objects along a closed curve for example a circle, the permutations are known as circular permutations. In a circular permutation, we have to consider one object as fixed and the remaining are arranged as in case of linear arrangement.

Linear arrangement is arrangement in a row.

Theorem
The number of circular permutations of n distinct objects is (n-1)!.

Anti-clock wise and clockwise order of arrangements are considered as distinct permutations in the above theorem.

If the anticlockwise and clockwise order is not distinct as in case of a garland which can be turned over easily, the number of distinct permutations will be ½ (n-1)!

Combinations and Difference between Combinations and Permutations

Each of the different selections made by taking some or all of a number of objects, irrespective of their arrangement is called a combination.


Difference between Combinations and Permutations

In a combination, the ordering of the selected objects is immaterial whereas in a permutation, the ordering is essential. For example AB and BA are same as combinations, but different as permutations.

Associate the word selection for combinations and arrangement for permuations.

Notation and Theorem for Combinations

Notation

The number of all combinations of n objects, taken r at a time is denoted by C(n,r) or ⁿC_r.

ⁿC_r is defined when n and r are non-negative numbers.

Theorem

The number of all combinations of n distinct objects, taken r at a time is given by

ⁿC_r = n!/(n-r)!r!

Results from the theorem

ⁿC_r = [n(n-1)(n-2)...(n-r+1)]/(1.2.3...r)

ⁿC_n =1

ⁿC₀ = 1


ⁿC_r = ⁿP_r/r!

Properties of C(n,r)

Properties of ⁿC_r or C(n,r)

1. ⁿC_r = ⁿC_n-r

Note: If x=y = n

ⁿC_x = ⁿC_y

2. Let n and r be non-negative integers such that r≤n. Then

ⁿC_r + ⁿC_r-1 = ⁿ⁺¹C_r

3. Let n and r be non-negative integers such that 1≤ r≤n. Then
ⁿC_r = (n/r) ^n-1C_r-1

4. If 1≤ r≤n, then

n.^n-1C_r-1 = (n-r+1)ⁿC_r-1

5. ⁿC_x = ⁿC_y implies x+y = n

6. If n is even, then the greatest value of ⁿC_r [0≤ r≤n] is ⁿC_n/2.

7. If n is odd, then the greatest value of ⁿC_r [0≤ r≤n] is ⁿC_(n+1)/2 or ⁿC_(n-1)/2.

Selection of one or more items

Selection from different items

The number of ways of selecting one or more items from a group of n distinct items is 2ⁿ - 1.

Selection from identical items

1. The number of ways of selecting r items out of n identical items is 1.
2. The total number of ways of selecting zero or more i.e. at least one item from a group of n identical items is (n+1).
3. The total number of selections of some or all out of p+q+r items where p are alike of one kind, q are alike of second kind, and rest are alike of third kind is {(p+1)(q+1)(r+1)}-1.

Selection of items from a group containing both identical and different items

1. The total number of ways of selecting one or more items from p identical items of one kind; q identical items of second kind, r identical items of third kind and n different items is

[(p+1)(q+1)(r+1) 2ⁿ]-1

Permutations and Combinations - Some Results

1. if n items are arranged in a row, then the number of wasys in which they can be rearranged so that no one of them occupies the place assigned to it (all wrongly placed), is

n![1 - 1/1! + 1/2! - 1/3! +...+(-1)ⁿ(1/n!)]

Pascal’s Triangle of Binomial coefficients

Expansions of (x+a)ⁿ as n = 0 to n


Each row is bounded by 1 on both sides
--- (The first row has only one item 1)First row is for n = 0
--- (The second row has two items 1 1)Second row is for n = 1
Any entry except the first and last entry in a row is the sum of two entries in the preceding row, one on the immediate left and the other on the immediate right.
So the third row (n = 2) is 1 2 1 (The second row elements are 1 1).

Binomial theorem

If x and a are real numbers, then for all n Є N

(x+a)ⁿ
= ⁿC₀xⁿa⁰ + ⁿC₁x^n-1a¹ +ⁿC₂x^n-2a² + ...+ⁿC_rx^n-ra^r+ ...+ⁿC_n-1x¹a^n-1+ⁿC_nx⁰aⁿ

(x+a)ⁿ
= (r = 0 to n)Σ ⁿC_rx^n-ra^r

Some Deductions from Binomial Theorem

1. total number of terms in the expansion = n+1

2. The sum of indices of x and a in each term is n.

3. the coefficients of terms equidistant from the beginning and the end are equal.

4. (x-a)ⁿ
= (r = 0 to n)Σ ((-1) ^r_*ⁿC_rx^n-ra^r

The terms in the expansion of(x-a)ⁿ are alternatively positive and negative, the last term is positive or negative according as n is even or odd.

5. (1+x) ⁿ = (r = 0 to n)Σ ⁿC_rx^r

you get it by putting x =1 and a = x in the expression for (x+a)ⁿ.

6. (x+1) ⁿ = (r = 0 to n)Σ ⁿC_rx^n-r

7. (1-x) ⁿ = (r = 0 to n)Σ(-1)^r_* ⁿC_rx^r

8. (x+a) ⁿ +(x-a) ⁿ = 2[ⁿC₀xⁿa⁰ +
ⁿC₂x^n-2a² +
ⁿC₄x^n-4a⁴+ ...]

9. General term in a binomial expansion

(r+1) term in (x+a) ⁿ
= ⁿC_rx^n-ra^r

Different version of Binomial theorem

(x+a)ⁿ = Σ (n!/r!s!) x^ra^s (where (r = 0 to n) and r+s = n)

Multinomial theorem

(x1+x2)ⁿ = Σ (n!/r1!r2!) x1^r1x2^r2 (where (r1 = 0 to n) and r1+r2 = n)


The result can be extended to (x1+x2_x3+…+xk) ⁿ

(x1+x2_x3+…+xk) ⁿ =
Σ [n!/(r1!r2!...rk!)] x1^r1x2^r2 xk^rk (where r1+r2+…+rn = n)

Middle terms in binomial expression

Binomial expression (1+x)ⁿ contains (n+1) terms when n is a natural number.

If n is even the ((n/2) +1) th term is middle term.
If n is odd then ((n+1)/2) th and ((n+3)/2)th terms are two middle terms

Results related to Binomial Coefficieints

1. Coefficient of (r+1)th term in the binomial expression of (1+x)ⁿ is ⁿC_r

2. Coefficient of x^r in the binomial expression of (1+x)ⁿ is ⁿC_r
3. 1. Coefficient of (r+1)th term in the binomial expression of (1-x)ⁿ is (-1)^rnC_r


4. Coefficient of x^r in the binomial expression of (1-x)ⁿ is (-1)^rnC_r

To find the greatest term in the expansion of (x+a)n

(i) Write (r+1)th term = T(r+1) and rth term = T (r)from the given expression.
(ii) Find T(r+1)/T(r)
(iii) Put T(r+1)/T(r)>1
(iv) Solve the inequality for r to get an inequality of the form rm

If m is an integer, the mth and (m+1)th terms are equal in magnitude and these two are the greatest terms.

If m is not an integer, then obtain integral part of m, say, k. In this case (k+1) term is the greatest term

Properties binomial coefficients

(i) the sum of binomial coefficients in the expansion of (1+x)ⁿ is 2ⁿ

(ii) the sum of odd binomial coefficients in the expansion of (1+x)ⁿ is equal to the sum of the coefficients of even terms and each is equal to 2^n-1.

iii) In the expansion of (1+x)ⁿ the coefficients of terms equidistant from the beginning and the end are equal.

(iv) C₀ – C₁ +C₂-C₃ +C₄-…+(-1) ⁿC_n = 0

Binomial theorem for any index (need not be a natural number)

If n is a rational number and x is a real number such that |x|<1, then

(1+x) ⁿ = 1 = nx + n(n-1)x²/2! + n(n-1)(n-2)x³/3!+…+n(n-1)(n-2)…(n-r+1)x^r/r!+...+ ∞

Remarks

The condition |x|<1 is unnecessary when n is a whole number.

When n is not a whole number, then the condition |x|<1 is necessary.

The terms are infinite when n is not an whole number. When is it an whole number the series become finite as one of the terms will become zero in the coefficient at some point in time.

When n is a positive integer, there will be n+1 terms

To expand (x+a)ⁿ proceed as follows:



(x+a)ⁿ = {a(1 + x/a)} ⁿ

= aⁿ(1 + x/a) ⁿ; (substitute x’ = x/a and proceed)

Number e and Related Ideas

e = lim(n→∞)(1 + 1/n)ⁿ

e lies between 2 and 3

e is an irrational number

Exponential Series

e = 1 +x/1! +x²/2!+x³/3!+...+∞

Exponential theorem

Let a is greater than 0.

for all real values of x,

a^x = 1 + x(log_ea) + (x²/2!)(log_ea)²+x³/3!(log_ea)³+...+∞

Relations of e (e-1, e-2 etc.)

n= 0 to ∞Σ1/n! = e

n= 1 to ∞Σ1/n! = e-1

n= 2 to ∞Σ1/n! = e-2

n= 0 to ∞Σ1/(n+1)! = e-1

n= 1 to ∞Σ1/(n+1)! = e

Matrix – Definition

Matrix is a set of mn numbers (real or imaginary) arranged in the form a rectangular array of m rows and n columns. It is called an m×n matrix and is read ‘m by b matrix’.




Concepts

1. A matrix is a rectangular array of numbers [a_ij]
2. A matrix with m rows and n columns is called an m×n matrix and the size or dimension of this matrix is said to be m×n.
3. Two matrices are said to be equal provided they are of the same dimension and corresponding elements of the two matrices are equal.
4. A matrix is termed as square matrix if m = n or its size is m×m.

Types of matrices

Row matrix
Column matrix
Diagonal matrix
Scalar matrix
Identity or unit matrix
Upper triangular matrix
Lower triangular matrix





5. A matrix is termed as row matrix if m = 1
6. A matrix is termed as column matrix if n = 1
7. A matrix is termed as null or zero matrix if a_ij = 0 for all i and j.
8. A matrix is termed as diagonal matrix if a_ij = 0 for all ij where i is not equal to j.
9. A matrix is termed as scalar matrix if a_ij = 0 for all ij where i is not equal to j and a_ij = constant (k) for all i and j.
10. A matrix is termed as identity matrix or unit matrix if a_ij = 0 for all ij where i is not equal to j and a_ij = 1 for all i and j.

Equality of matrices

Two matrices

A is an m × n matrix [a_ij]

B is an r × s matrix [b_ij]

A and B are equal if

i. m= r (number of rows are same)
ii. n = s (number of columns are same)
iii. a_ij = b_ij for I =1,2,…,m abd j = 1,2,…,n

Algebra of Matrices

Addition of matrices

If two matrices are of the same order m x n, then their sum is a matrix of order m x n and is obtained by adding the corresponding elements.

(A+B)_ij = a_ij + b_ij

Multiplication of a matrix by a scalar

kA = [ka_ij]_{m x n}

Subtraction of matrices

A – B = A +(-B)

Multiplication of matrices

Two matrices A and B are conformable for multiplication if the number of columns in A is same as the number of rows in B.

A is premultiplier matrix and B is called post multiplier matrix.

AB is defined in the following way.

(AB) _ij = Σ(r= 1 to n) a_irb_rj
= a_i1b_1j+a_i2b_2j+…a_imb_mj

When AB exists, BA may or may not exist.

Properties of matrix multiplication

1. Matrix multiplication is not commutative in general.
2. Matrix multiplication is associative.
3. Matrix multiplication is distributive over matrix.
4. If A is an m x n matrix, then I_mA = A = AI_n
5. The product of two matrices can be the null matrix while neither of them is the null matrix.
6. Product of the matrix with a null matrix is always a null matrix.
7. If AB = 0 when either of them is not zero, it does not imply BA is zero.


Positive integral powers of A
Aⁿ⁺¹ = AⁿA
(A^m)ⁿ = (A^-)mn


Matrix polynomial = a₀Aⁿ+a₁A^n-1+…+a_n-1A+a_nI_n.

Traspose of a matrix

Tranpose of a matrix A^T is obtained from A by changing its rows into columns and its columns into rows.
The first row of A is the first column of A^T.

Properties of Transpose

1. (A^T)^T = A
2. (A+B) ^T = A^T+B^T ( A and B must have the same order)
3. (kA) ^T = kA^T., (k is any scalar)
4. (AB) ^T = B^TA^T

Symmetric and skew symmetric matrices

Symmetric matrix

A square matrix is called a symmetric matrix iff a_ij = a_ji for all I,j.
It means (A)_ij = (A^T) _ij

skew symmetric matrix
A square matrix is called a skew-symmetric matrix iff a_ij = -a_ji for all I,j.
It means (A)_ij = -(A^T) _ij
It means A^T = -A

Determinant of a matrix

Every square matrix can be associated to an expression or a number which is known as determinant.

If the matrix has only one element a₁₁ then a₁₁ is the determinant.

If the matrix is of order 2 that 2 by 2 matrix

|A| =

|a11 a12|
|a21 a22| =

a11*a22 – a12*a21

Determinant of a square matrix of order 3

Determinant of a square matrix of order 3 is the sum of the product of elements a1j in the first row with (-1) ^1+j times the determinant of a 2×2 sub-matrix obtained by leaving the first row and column passing through the element.



(i) Only square matrices have determinants.
(ii) The determinant of a square matrix of order three can be expanded along any row or column.

Determinant of a square matrix of order 4 or more

(iii) Determinant of a square matrix of order 4 or more can be determined following the procedure of finding the determinant of a square matrix of order 3. But in this case, especially in the case of 4×4 matrix, when we omit the rows and columns containing the elements of a row, we get 3×3 sub-matrices and we have to find determinants for them.

Singular Matrix

A square matrix is a singular matrix if its determinant is zero

Inverse of a matrix

Let A be a square matrix of order n

If AB = I_n = BA

The B is inverse of A and is written as
A^-1 = B



Theorems related to Inverses of matrices

1. Every invertible matrix possesses a unique inverse

2. A square matrix is invertible iff it is nonsingular.

3. A^-1 = (1/|A|)adj A

4. Cancellation laws: Let A, B, and C be square matrices of the same order n. If A is a non-singular matrix, then

(i) AB = AC => B = C … (left cancellation law)
(ii) BA = CA => B = C … (right cancellation law)

This law is true only when |A| ≠ 0. Otherwise, there may be matrices such that AB = AC but B≠C.

5. Reversal law: If A and B are invertible matrices of the same order, then AB is invertible and

(AB) ^-1 = B^-1A^-1

6.If A,B,C are invertible matrices then
(ABC) ^-1 = C^-1B^-1A^-1

7.If A is an invertible square matrix, then A^T is also invertible and
(A^T)^-1 = (A^-1)^T

8. Let A be a non-singular square matrix of order n. Then

|adj A| = |A|^n-1

9. If A and B are non-singular square matrices of the same order, then

adj AB = (adj B) (adj A)

10. If A is an invertible square matrix, then

adj A^T = (adj A) ^T


11. If A is a non-singular square matrix, then

adj(adj A) = |A|^n-2A

Elementary Transformations or Elementary Operations of a Matrix

1. Interchange of two rows or columns.
2. Multiplication of all elements of a row or column of a matrix by a non-zero scalar,
3. Addition to the elements of a row or column of the corresponding elements of any other row (to a row) or any other column (to a column) multiplied by a scalar k.


Elementary matrix: A matrix obtained from an identity matrix by a single elementary operation (transformation) is called an elementary matrix.

Orthogonal matrix

A square matrix is called an orthogonal matrix if AA^T = A^TA = I

Submatrix

From an m x n matrix, other matrices obtained by leaving some rows or columns are called submatrices of the m x n matrix.

Rank of a Matrix

A number r is said to be the rank of a an "m x n" marix if
i) every square sub matrix of it of order (r+1) or more is singular, and
ii) there exists at least on square matrix of order r which is non-singular.

In other words, the rank of a m x n matrix is the order of the highest order non-singular square submatrix of it.

Some Theorems on Rank of a Matrix

The rank of matrix is the same as that of its transpose.

Elementary transformations do not alter the rank of a matrix

Equivalent Matrices

Two matrices of equivalent if one can be obtained from the other by a sequence of elementary row transformations.

Echelon Form of a Matrix

i) Every non-zero row precedes every zero row in the matrix.
ii) The number of zeros before the first non-zero element in a row is less than the number of such zeros in the next row.

Rank of a matrix in echelon form is equal to the number of non-zero rows of the matrix

System of simultaneous Linear Equations

It is written as AX = B

The m x n matrix A is called the coefficient matrix of the system of linear equations

If B = 0, the system is called homogeneous system.
Otherwise, it is non-homogeneous system.

Solution of a Non-Homogeneous System of Linear Equations

Three methods are there

1. Matrix method
X is obtained by finding out A^-1B

2. Determinant method – Cramer’s rule

3. Rank Method

If |A| ≠0, then the system is consistent and has a unique solution A^-1B.
If |A| = 0, and (adj A)B = 0, then the system is consistent and has infinitely many solutions.

If |A| = 0, and (adj A)B ≠ 0, then the system is inconsistent and has no solution.

Rank Method of solving System of Linear Equations

For the system of equations

AX = B

Write [A:B] by adding the column of B to A.

[A:B] is called the augmented matrix for the given system of equations.

1. If the rank of augmented matrix is equal to rank of the coefficient matrix, the given system is consistent.

2a. In case of m x n matrix, if m is greater than and r(A) = r(A:B) = n, the system has a unique solution.
2b. In case of m x n matrix, if m is greater than and r(A) = r(A:B) is less than n, the system has multiple (infinite) solutions
2c. Similarly, In case of m x n matrix, if m is less than and r(A) = r(A:B) is less than m, the system has multiple (infinite) solutions

Solution of a Homogeneous System of Linear Equations

AX = 0

In this case if A is nonsingular. X = 0 is the only solution. It is a trivial solution.

A system AX = 0 of n equations in n unknowns has non trivial solution iff the coefficieint matrix is singular.

To solve the system with singular A in 3 variables (as an example) Put z =k and solve the remaining equations for x and y in terms of k. This is the general solution to the problem.

A homogeneous system of linear equations is always consistent.

Determinants - Definition

Every square matrix can be associated to an expression or a number which is known as determinant.

If the matrix has only one element a11 then a11 is the determinant.

If the matrix is of order 2 that 2 by 2 matrix

|A| =

|a11 a12|
|a21 a22| =

a11*a22 – a12*a21

Determinant of a square matrix of order 3

Determinant of a square matrix of order 3 is the sum of the product of elements a1j in the first row with (-1) 1+j times the determinant of a 2×2 sub-matrix obtained by leaving the first row and column passing through the element.



(i) Only square matrices have determinants.
(ii) The determinant of a square matrix of order three can be expanded along any row or column.

Determinant of a square matrix of order 4 or more

(iii) Determinant of a square matrix of order 4 or more can be determined following the procedure of finding the determinant of a square matrix of order 3. But in this case, especially in the case of 4×4 matrix, when we omit the rows and columns containing the elements of a row, we get 3×3 sub-matrices and we have to find determinants for them.

Singular matrix

A square matrix is a singular matrix if its determinant is zero.
Otherwise it is a non-singular matrix.

Minors and cofactors

Minor: For a square matrix [aij] or order n, the minor Mij of aij, in A is the determinant of the square sub-matrix of order (n-1), obtained by leaving (or striking off) ith row and jth column of A.

Cofactor: Cofactor of an element aij in a square matrix [aij] is termed Cij.

C_ij = (-1)^i+jM)_ij

M_ij is the minor of element a_ij in a square matrix [a_ij].

Minors and cofactors are defined for elements of a square matrix only. They are not defined for determinants.

Properties of determinants

1. For a square matrix, the sum of the product of elements of any row (or column) with their cofactors is always equal to determinants of the matrix.

2. For a square matrix, the sum of the product of elements of any row (or column) with the cofactors of corresponding elements of some other row (or column) is zero.

3. The value of a determinant remains unchanged if its rows and columns are interchanged.

4. If any two rows (or columns) of a determinant are interchanged, then the value of the determinant changes by minus sign only.

5. If any two rows or columns of a determinant are identical then its value is zero.

6. If each element of a row (or a column) of a determinant is multiplied by a constant k, then the value of the new determinant is ‘k’ times the value of the original determinant.

7. If each element of a row (or a column) of a determinant is expressed as a sum of two or more terms, then the determinant can be expressed as the sum of two or more determinants of the same order.

8. If each element of a row (or a column) of a determinant is multiplied by the same constant and then added to the corresponding elements of some other row (column) then the value of the determinant remains same.

9. If each element of a row (or column) in a determinant is zero, then its value is zero.

10. If the matrix is a diagonal square matrix then its determinant is the product of all the diagonal elements.

11. If A and B are square matrices of the same order, then

|AB| = |A| |B|

12. If a matrix is a triangular matrix of order n, then its determinant is the product of all the diagonal elements.

Evaluation of Determinants

To evaluate determinants or large matrices, we use the properties of determinants, to create many zeroes in the elements of a row or column and then expand the determinant using elements and cofactors of that row or column.

Evaluation of Determinants by using Factor Theorem

If f(x) is a polynomial and f(α) = 0 the, (x- α) is a factor of f(x).

If a determinant is a polynomial in x, then (x- α) is factor of the determinant if its value is zero when we put x = α.

Using this rule we can find determinant as a product of its factors.

Product of Determinants

A definition of product of determinants is similar to the rule of multiplication of matrices.

Differentiation of Determinants

If Δ(x) =

|f1(x) g1(x)|
|f2(x) g2(x)|

its differentiatin Δ'(x) =

|f1'(x) g1'(x)|
|f2(x) g2(x)|

Plus
|f1(x) g1(x)|
|f2'(x) g2'(x)|

This means to differentiate a determinant, we differentiate one row at a time, keeping others unchanged and then add them

If there are two rows R1 and R2

Δ'(x) =

|R1'|
|R2|

Plus
|R1|
|R2'|

Euclidean Geometry and Analytic Geometry

Euclidean geometry: In it one starts with certain concepts and axioms and use the methods of deductive logic to derive a number of theorems and results. These help us know a number of useful properties of geometrical figures.

Analytical geometry or coordinate geometry: In this, points in a plane are represented by ordered pairs of real numbers, called cartesian coordinates and lines and curves are represented by algebraic equations.

Distance between two points

P(x1,y1) Q(x2,y2)

PQ distance = √[(x2-x1)² +(y2-y1)²]

Area of a triangle

(1/2)[x1(y2-y3)+x2(y3-y1)+x3(y1-y2)]

Section Formulae

Formula for internal division
A(x1,y1), B(x2,y2) AB in ratio m:n (AC:CB)

x = (mx2 + nx1)/(m+n); y = (my2 + ny1)/(m+n)

Formula for external division

x = (mx2 - nx1)/(m+n); y = (my2 - ny1)/(m+n)

coordinates of the centres related to triangle

centroid ((x1+x2+x3)/3, (y1+y2+y3)/3)

In-centre

A(x1,y1)

B(x2,y2)

C(x3,y3)

Incentre ((ax1+bx2+cx3)/(a+b+c), (ay1+by2+cy3)/(a+b+c))

Ex centres

centre of the escribed circle opposite to the angle A

((-ax1+bx2+cx3)/(-a+b+c), (-ay1+by2+cy3)/(-a+b+c))

Definition of a straight line

A straight line is a curve such that every point on the line segment joining any two points on it lies on it.

every first degree equation in x,y represents a straight line.

Slope (Gradient) of a straight line

The general equation of a line is of the form ax +by +c = o and its slope is –a/b, provided b≠0.

Anlge between two straight lines

If m1 and m2 are the slopes of two lines, then the acute angle θ between them is given by tan θ = |m1-m2|/|1 + m1*m2|, provided m1*m2≠-1.

Intercepts of a line on the axes

If a straight line cuts x axis at A and O is origin AO or OA is the x-intercept.

If a straight line cuts y axis at B and O is origin BO or OB is the y-intercept.

Two lines - Point of intersection

Important properties and centres related to triangle

Centroid

Incentre

Circum-centre

Orthocentre

Joint equation of a pair of straight lines

The joint equation of a family of lines is always of second degree in x and y.

The joint equation of the straight lines a1x +b1y +c1 = 0 and a2x +b2y+ c2 = 0 is
(a1x +b1y +c1) (a2x +b2y+ c2) = 0

The equation ax² +2hxy+by² = 0 is known as homogeneous equation of second degree.
In a homogeneous equation of second degree, the sum of indices (exponents) of x and y in each term is equal to 2.

The homogeneous equation of second degree ax² +2hxy+by² = 0 represents a joint equation of two straight lines passing through the origin if h²≥ab.

If y = m1x and y = m2x are the lines represented by a homogeneous equation of second degree ax² +2hxy+by² = 0, then

(i) m1 =m2 = -2h/b
(ii) m1m2 = a/b

Angle between pair of straightlines represented by homogeneous equation

General equation of second degree for a pair of straight lines.

The equation ax² +2hxy+by²+2gx+2fy+c = 0 is known as general equation of second degree.

The equation ax² +2hxy+by² = 0 is known as homogeneous equation of second degree.
In a homogeneous equation of second degree, the sum of indices (exponents) of x and y in each term is equal to 2.

The homogeneous equation of second degree ax² +2hxy+by² = 0 represents a joint equation of two straight lines passing through the origin if h²≥ab.

If y = m1x and y = m2x are the lines represented by a homogeneous equation of second degree ax² +2hxy+by² = 0, then

(i) m1 =m2 = -2h/b
(ii) m1m2 = a/b

The angle θ between the pair of lines represented the homogeneous equation of second degree ax² +2hxy+by² = 0 is given by

tan θ = [2√(h² –ab)]/(a+b)

If θ = 0, which means h² = ab lines are coincident.

Lines are perpendicular means θ = π/2, tan θ = ∞, and cot θ = 0.
This means a+b = 0 or a = -b
Coefficient of x² = coefficient of y²

ax² +2hxy+by²+2gx+2fy+c = 0 will represent a pair of straight lines if the determinant


|a h g|
|h b f|
|g f c|

= 0

Expanding the determinant

abc +2fgh -af² -bg² -ch² = 0


Angle θ between the lines represented by the general second degree equation ax² +2hxy+by²+2gx+2fy+c = 0 is given by

tan θ = [2 √(h² – ab)]/(a+b)



Algorithm to find separate equations of lines in ax² +2hxy+by²+2gx+2fy+c = 0

Step 1. Find factors for the homogeneous part ax² +2hxy+by². Let the factors be
(a1x +b1y ) and (a2x +b2y )

Step 2.Add constants c1 and c2 to them. (a1x +b1y +c1) and (a2x +b2y +c2).

Step 3. Multiply (a1x +b1y +c1) and (a2x +b2y +c2) and compare with ax² +2hxy+by²+2gx+2fy+c to obtain equations in c1 and c2.

Step 4. Solve the equations and get values of c1 and c2.

Equations for bisectors of the lines represented by ax² +2hxy+by²+2gx+2fy+c = 0

[(x-x1) ² – (y-y1) ²]/(a-b) = (x-x1)(y-y1)/h
where x1,y1 is the point of intersection of the lines represented by the given equation.

Lines joining the origin to the points of intersection of a line and a curve.

Step 1. Take all terms of x and y in the equation of the line on LHS and the constant term on RHS, then divide both sides by the this constant on RHS, so that RHS becomes unity.

Step 2. Multiply the first degree terms in the equation of the curve by the LHS obtained in step 1 and the constant term by the square of the LHS obtained in the step 1, keeping the second degree terms unchanged. The required equation is obtained.

Cricle - Definitions

A circle is defined as the locus of a point which moves in a plane such that its distance form a fixed point in that plane is always fixed.

Intercept of the circle on x axis is the length of chord of the circle which is a part of x axis.

Similarly Intercept of the circle on y axis is the length of chord of the circle which is a part of y axis.

Director circle: the locus of the point of intersection of two perpendicular tangents to a given conic is known as its director circle.

Chord of contact: the chord joining the points of contact of the two tangents to a conic drawn from a given point, outside it, is called the chord of contact of tangents.

Pole and Polar:
Polar of a point with respect to a circle: Of through a point P(x1,y1) (inside or outside a circle) there be drawn any straight line to meet the given circle a Q and R, the locus of the point of intersection of the tangents at Q and R is called the polar of point P and P is the called the pole of the polar.

Polar is the locus and pole is a point.

Diameter – definition as a locus: the locus of the middle points of a system of parallel chords of a circle is called a diameter of the circle.

Common chord of two circles: The chord joining the points of intersection of two given circles is called their common chord.

Angle of intersection of two curves: If the two curves C1 and C2 intersect at a point P and PT1 and PT2 be the tangents to the two curves C1 and C2 respectively at P. Then the angle between the tangents at P is called the angle of intersection of the two curves at the point of intersection.

Orthogonal curves: Two curves are said to intersect orthogonally when the two tangents at the common point are at right angles.

Radical axis: the radical axis of two circles is the locus of a point which moves in such a way that the lengths of the tangents drawn from it to the two circles are equal.

Radical centre: The point of concurrence of the radical axes of three circles whose centres are non-collinear, taken in pairs, is called the radical centre of the circles.

Coaxial system of circles: A system of circles, every pair of which has the same radical axis is called a coaxial system of circles.

Standard equation of a circle

(x-h)²+(y-k)² = a²

Centre of the circle is at (h,k)
radius of the circule is a

Some particular cases of standard equation of a circle

i) Centre is at origin h = 0, and k = 0

x²+y² = a²

(ii) Circle passes through origin
So radius = a² = h²+k²

(x-h)²+(y-k)² = h²+k²

(iii)Circle touches the x axis
C(h,k) centre, a = radius
To satisfy a = k
So equation is
(x-h)²+(y-a)² = a²

(iv)Circle touches the y axis
C(h,k) centre, a = radius
To satisfy a = h
So equation is
(x-a)²+(y-k)² = a²

(v) When the circle touches both axes

then h = k = a
(x-a)²+(y-a)² = a²

(vi) When the circle passes through the origin and centre is on x-axis.
C(h,k) centre, a = radius

As centre is on x axis y coordinate is zero. So k = 0.
As circle is passing through origin a = h
(x-a)²+ y² = a²

(vii) When the circle passes through the origin and centre is on y-axis.
C(h,k) centre, a = radius

As centre is on y axis x coordinate is zero. So h = 0.
As circle is passing through origin a = k
x²+(y-a)² = a²

General equation of a circle

x²+y²+2gx+2fy+c = 0

Centre of this circle = (-g,-f)
Radius = √(g²+f²-c)

Equation of a circle when the coordinates of end points of a diameter are given

If (x1,y1) and (x2,y2) are coordinates of end points of the diameter

then the equation of the circle is
(x - x1)(x - x2)+(y - y1)(y- y2) = o

Intercepts of the axes

Intercept of a circle is a line that is a chord which is part of x axis

Intercepts for the circle x²+y²+2gx+2fy+c = 0

length of intercept on x- axis = 2√(g²-c)(You get it by putting y = 0)
length of intercept on y- axis = 2√(f²-c)(You get it by putting x = 0)

Position of a point with respect to a circle

Is a point in the circle, on the circle or outside the circle

If the point is P find distance between the centre of the circle C and point P.
If the radius of the circle be R

CP is greater than R implies point is outside the circle.

CP = R implies point is on the circle

CP is less than R implies point is inside.

Equation of a circle in parametric form

Parametric equations of x² + y² = r²

x = r cos θ, y = r sin θ

Parametric equations of (x-a)² + (y-b)² = r²

x = a + r cos θ, y = b + r sin θ

Intersection of a straight line and a circle

Equation of the circle: x² + y² = a²

Equation of the line: y = mx+c

A line does not intersect a circle if the length of the perpendicular to the line from the centre of the circle is greater than the radius of the circle.
|c/√(1+m²)|>a

A line intersects a circle if the length of the perpendicular to the line from the centre of the circle is less than the radius of the circle.

|c/√(1+m²)|
A line touches a circle if the length of the perpendicular to the line from the centre of the circle is equal to the radius of the circle.

|c/√(1+m²)| = a

The length of the intercept cut off from a line by a circle

Equation of the circle: x² + y² = a²

Equation of the line: y = mx+c

A line intersects a circle if the length of the perpendicular to the line from the centre of the circle is less than the radius of the circle.

If it intercepts, the length of the intercept is

2√([[a²(1+m²)-c²]/(1+m²) ]

Tangent to a circle at a given point

Condition of tangency:

The line y = mx+c is tangent to a circle x² + y² = a² if the length of the intercept is zero.
That means 2√([[a²(1+m²)-c²]/(1+m²) ] = 0
=> a²(1+m²)-c² = 0
=> c = ±a√(1+m²)


Slope form:

The equation of a tangent of slope m to the circle x² + y² = a² is
Y = mx±a√(1+m²) (Value of c from tangent condition).
The coordinate of the point of contact are (±am/√(1+m²), - or +a/√(1+m²)


Point form:

The equation of a tangent at the point (x1,y1) to the circle x² + y²+2gx+2fy+c = 0 is

xx1 + yy1 +g(x+x1)+f(y+y1) +c = 0

Normal to a circle at a given point

If slope of the tangent is m, then the slope of the normal is –(1/m)

Length of the tangent from a point to a circle

The length of a tangent from the point (x1,y1) to the circle x² + y²+2gx+2fy+c = 0 is equal to √( x1² + y1²+2gx1+2fy1+c)

Pair of tangents drawn from a point to given circle

Let the point be (x1,y1) and the circle be x² + y² = a²

Two tangents can be drawn.

The tangent will be of the form y = mx+a√(1+m²)
And the two values of m for the pair is to be found by solving the quadratic equation
m²(x1²-a²) -2mx1y1 +(y1²-a²) = 0

Combined equation of pair of tangents drawn from a point (x1,y1) to a circle

The equation for pair of tangents from the point (x1,y1) to the circle x² + y²+2gx+2fy+c = 0 is given by

(x² + y²+2gx+2fy+c) (x1² + y1²+2gx1+2fy1+c) = (xx1 + yy1 +g(x+x1)+f(y+y1) +c) ²

Expressed as SS’ = T²

Director circle of a circle and its equation

Director circle: the locus of the point of intersection of two perpendicular tangents to a given conic is known as its director circle.


Equation of director circle of the circle x² + y² = a² is x² + y² = 2a²

Chord of contacts of tangents of a circle

The equation of the chord of contact of tangents drawn from a point (x1,y1) outside the circle to the circle x² + y² = a² is xx1+yy1 = a².

Pole and Polar of a point with respect to a circle

Polar of a point with respect to a circle: If through a point P(x1,y1) (inside or outside a circle) there be drawn any straight line to meet the given circle a Q and R, the locus of the point of intersection (T) of the tangents at Q and R is called the polar of point P and P is the called the pole of the polar.

Polar is the locus of point and pole is a point with respect to which polar is determined.



Equation to the polar of the point (x1,y1) w.r.t. to the circle x² + y² = a² is

xx1+yy1 = a²

The polar of the point (x1,y1) w.r.t. to the circle x² + y²+2gx+2fy+c = 0 is given by
(xx1 + yy1 +g(x+x1)+f(y+y1) +c) = 0
The equation is same as the equation for the tangent to the circle at a point (x1,y1) on the circle.

Equation of the chord bisected at a given point

The equation of the chord of the circle x² + y²+2gx+2fy+c = 0 bisected at the point (x1,y1) is given by

T = S’
(xx1 + yy1 +g(x+x1)+f(y+y1) +c) = x1² + y1²+2gx1+2fy1+c

Diameter of a circle – Locus of middle points of parallel chords

Equation of the diameter bisecting parallel chords y =mx+c ( c is a parameter i.e., varies to give various chords) of the circle x² + y² = a² is x+my = 0

Common tangents to two circles

Let the two circles be

(x-h1)² + (y-k1)² = a²

(x-h2)² + (y-k2)² = a²

with centres C1(h1,k1) and C2(h2,k2) and radii a1 and a2 respectively.

The various cases that can occur are

Case 1. When C1C2>a1+a2 i.e., the distance between the centres is greater than the sum of radii.
In this case, the circles do not intersect each other and four common tangents can be drawn to two circles.
Two of them are direct common tangents. Two are transverse common tangents.
The intersection between common tangents (T2) lies on the line joining C1 and C2 and divides it externally in the ratio a1/a2. C1T2/CTs = a1/a2

The intersection between transverse tangents (T1) lies on the line joining C1 and C2 and divides the line internally in the ration a1/a2. i.e., C1T1/C2T1 = a1/a2.

Case 2. When C1C2 = a1+a2 i.e, the distance between the centres of circles is equal to the sum of the radii, two direct tangents are real and distinct, but the transverse tangents are coincident.

Case 3. When C1C2
Case 4. When C1C2 = a1-a2 i.e., the distance between the centres is equal to the difference of the radii.

In this case two tangents are real and coincident while the other two tangents are imaginary.

Case 5. When C1C2 < a1-a2 i.e., the distance between the centres is less than the difference of the radii.

In this case all four common tangents are imaginary.

Common chord of two circles

Equation
2x(g1-g2)+2y(f1-f2)+c1-c2 = 0

This is for circles

Circle 1 (termed as S1) x² + y²+2g1x+2f1y+c = 0

Circle 2 (termed as S2) x² + y²+2g2x+2f2y+c = 0

Length of the common chord = 2√ (C1P²-C1M²)
Where
C1P = radius of circle 1
C1M = length of the perpendicular from the centre C1 to the common chord PQ.

Angle of intersection of two curves and the condition of orthogonality of two circles

Condition for two intersecting circles to be orthogonal

Let Circle 1 (termed as S1) be x² + y²+2g1x+2f1y+c = 0

And Circle 2 (termed as S2) be x² + y²+2g2x+2f2y+c = 0

Condition is 2(g1g2+f1f2) = c1+c2

Radical axis of two circles

For two circles

Circle 1 (termed as S1) x² + y²+2g1x+2f1y+c = 0

Circle 2 (termed as S2) x² + y²+2g2x+2f2y+c = 0

Radical axis is
S1-S2 = 0

2x(g1-g2)+2y(f1-f2)+c1-c2 = 0

The equation has the same form at that of common chord of intersecting circles.

Properties of radical axis:
(i) The radical axis of two circles is always perpendicular to the line joining the centres.
(ii) The radical axes of three circles whose centres are non-collinear, taken in pairs, meet in a point. (This point is called radical centre)
(iii) The circle with centre at the radical centre and radius equal to the length of the tangent from it to any of the circles intersects all three circles orthogonally.

Equation of a circle through the intersection of a circle and line

The equation of a circle passing through the intersection (points of intersection) of the circle S = x² + y²+2g1x+2f1y+c = 0 and the line L = lx+my+n = 0 is
x² + y²+2g1x+2f1y+c+ λ(lx+my+n) = 0 or
S+ λL = 0 where λ is a constant determined by an additional condition.

Circle through the intersection of the two circles

The equation of a family of circles passing through the intersection of the circles

Circle 1 (termed as S1) x² + y²+2g1x+2f1y+c = 0

Circle 2 (termed as S2) x² + y²+2g2x+2f2y+c = 0

Is S1+ λS2 = 0

Coaxial system of circles

The equation x² + y²+2gx+c = 0, where g is a variable and c is a constant is the simplest equation of a coaxial system of circles. The common radial adxis of this system of circles is y-axis.

If the equation of one of the circles and the radical axis are given:
Circle x² + y²+2gx+2fy+c = 0
Radical axis P = lx+my+n = 0

Then S+ λP = 0 (λ is an arbitrary constant) represents the coaxial system of circles.

If the equations of two of the circles are given

Circle 1 (termed as S1) x² + y²+2g1x+2f1y+c = 0

Circle 2 (termed as S2) x² + y²+2g2x+2f2y+c = 0

Then S1+λS2 = ) (λ ≠-1) represents the coaxial system.

The parabola - definitions

Parabola is the locus of a point P which moves in a plane so that its distance from a fixed line of the plane and its distance from a fixed point of the plane, not on the line, are equal.

The fixed point F is called the focus and fixed line is called the directrix of the parabola.

The perpendicular to he directix from the focus is called the axis of the parabola.

The intersection of the parabola and the axis of the parabola is called vertex.

Vertex is the mid point of axis.

The line joining any two distinct points of the parabola is called a chord.

A chord which passes through the focus is called a focal chord.

The distance between the focus and any point of the focal chord is called focal radius.

The focal chord which is penpendicular to the axis is called the latus rectum

Equation of parabola in its standard form

y² = 4ax

For this equation focus is at F(a,0) and the equation of the directrix is d: x=-a. It vertex is at (0,0).

If a is positive it open to the right.

Length of the latus rectum = |4p|

Some other standard forms of parabola

y² = -4ax

x² = 4ay

In this case, the vertex is at the origin and the axis coincides with y-axis.
Focus is at F(0,a) and the equation of the directrix d: y = -a.
The parabola opens upward

x² = -4ay

Equation of a parabola in parametric form

x = at²
y = 2at


It satisfies y² = 4ax
y² = 4a²t²
4ax = 4a²t²

Equation of the chord joining any two points on the parabola

From the straight line chapter we know: "The equation of a line having slope m and passing through (x1,y1) is

(y-y1) = m(x-x1)"

slope between (x1,y1) and (x2,y2) = (y2-y1)/(x2-x1)

Two points on parabola are A(at1²,2at1) and B(at2²,2at2)

So the equation joining these two points is

(y-2at1) = [(2at2-2at1)/(at2²-at1²)]*(x-at1²)
=> y - 2at1 = [2/(t2+ta)]*(x-at1²)
=> y(t1+t2) = 2x+2at1t2

Intersection of a straight line and a parabola

Parabola equation y² = 4ax,
Straight line equation y = mx+c

At intersection point, both equations are satisfied

hence (mx+c)² = 4ax
=> m²x²+2x(mc-2a)+c² = 0

It is a quadratic equation. Solution gives intersection points

The intersection points are concident if
4(mc-2a)² - 4m²c²>0
=> a² - amc>0
=>a-mc>0
=>a>mc
=>a/m>c
=>c
The intersection points are real and distinct if
4(mc-2a)² - 4m²c²=0
=> 4(mc-2a)² = 4m²c²

The intersection points are imaginary if
4(mc-2a)² - 4m²c²<0

Equation of tangent to a parabola in different forms

For the Parabola with equation y² = 4ax;

Point form: At point (x1,y1)
(y-y1) = (2a/y1)*(x-x1)
=> yy1 = 2a(x+x1)

In parametric form Points is (at²,2at)
ty = x+at²

In terms of slope of the tangent, If slope is m
y = mx + a/m

Equation of normal in different forms

Parabola equation y² = 4ax; at point (x1,y1)
(y-y1) = (-y1/2a)* (x-x1)

In parametric form Points is (at²,2at)
y+tx = 2at + at³

Slope form, slope of the normal = m
y = mx-2am-am³

Number of normals drawn from a point to a parabola

In general three normals can be drawn from a point to a parabola

Conormal points

Conormal points : The points on the curve at which hte normals pass through a common point are called co-normal points.

The sum of the slopes of the normals at conormal points is zero.

Number of tangents drawn from a point to a parabola

Two tangents can be drawn from a point to a parabola

Equation of the pair of tangents from a point to a parabola

Parabola equation y² = 4ax; point from which tangents are drawn is (x1,y1)

Equation is SS' = T²
S = y² = 4ax
S' = y1² = 4ax1
T = yy1 - 2a(x+x1)

Equation of the chord of contacts of tangents to a parabola

From a point two tangents are drawn to a parabola. The chord between the contact points of these two tangents is chord of contact of tangents.

When Parabola equation y² = 4ax; point from which tangents are drawn is (x1,y1)

chord equation is yy1 = 2a(x+x1)

Equation of the chord bisected at a given point

When a chord to Parabola equation y² = 4ax is bisected at (x1,y1)

equation of the chord is yy1 - 2a(x+x1) = y1²-4ax1

Equation of diameter of a parabola

The locus of bisectors of a system of parallel chords is termed diameter.

If Parabola equation is y² = 4ax, and system of parallel chords equation is y = mx+c,

The equation of the diameter is y = 2a/m

It is a line parallel to the X-axis.

Lengths of tangent, subtangent, normal and subnormal

Let tangent and normal to a parabola at a point P(x1,y1) be extended to meet the axis of the parabola at N and T respectively.

PT is termed the length of the tangent.
PN is termed the length of the normal.

Drop a perpendicular to the axis from the point P and call it PP'.

P'T = subtangent
P'N = subnormal

If the tangent makes an angle of ψ with the axis

length of the tangent = y1 cosec ψ
length of the normal = y1 sec ψ
length of the subtangent = y1 cot ψ
length of the subnormal = y1 tan ψ

tan ψ = 2a/y1 = m (slope of the tangent)

Equation of ellipse in its standard form

x²/a² + y²/b² = 1


In x²/a² + y²/b² = 1, if a>b or a²>b² (denominator of x² is greater than that of y²), then the major and minor axes lie along x-axis and y-axis respectively.

The centre of the ellipse will be at (0,0).

Coordinates of the vertices will be at (a,0) and (-a,0).

Length of the major axis is 2a.
Length of the minor axis is 2b

Equation of major axis y = 0
Equation of minor axis x = 0

Eccentricity e = √(1 - b²/a²)

Length of latus rectum = 2b²/a

Equations of directrices x = a/e and x = -a/e

Equation of ellipse in other forms

If the centre of the ellipse is at point (h,k) and the directions of axes are parallel to the coordinate axes, then its equation is

(x-h) ²/a² + (y-k) ²/b² = 1

Position of a point with respect to an ellipse

The point p(x1,y1) lies outside, on or inside the ellipse x²/a² + y²/b² = 1
According as x1²/a² + y1²/b²- 1>, = or ,0.

Parametric equations and parametric coordinates of an ellipse

The circle described on the major axis of an ellipse as diameter is called an auxiliary circle of the ellipse.

For the ellipse x²/a² + y²/b² = 1
Auxiliary circle is x² + y² = a²

Parametric coordinates of ellipse

If ф is the angle made by the line joining centre of ellipse with a point (x,y) on the ellipse x = a cos ф and y = b sin ф are the parametric coordinates of the ellipse x²/a² + y²/b² = 1, a>b

Equation of the chord joining any two points on an ellipse

If P(a cos θ, b sin θ) and Q( a cos ф,b sin ф) be any two points of the ellipse x²/a² + y²/b² = 1, then equation of chord joining P and Q is of the form

(y-y1) = m(x-x1) which is going to be

(y – b sin θ) = [(b sin ф - b sin θ)/( a cos ф - a cos θ)]*(x – a cos θ)

The equation can also be expressed as

(x/a)* cos [(θ+ ф)/2] +(y/b)*sin [(θ+ ф)/2] = cos [(θ- ф)/2]

Condition of a line to be a tangent to an ellipse

The condition for the line y = mx+c to be a tangent to the ellipse x²/a² + y²/b² = 1 is that c² = a²m²+b² or c = ±√( a²m²+b²)

Hyperbola – Definition

A hyperbola is the locus of a point in a plane which moves in the plane in such a way that the ratio of its distance from a fixed point (called focus) in the same plane to its distance from a fixed line (called directrix) is always constant which is always greater than unity.

The constant ratio is generally denoted by ‘e’ and is known as the eccentricity of the hyperbola.


Every hyperbola has a second focus and second directrix.

The difference of the focal distances of any point on a hyperbola is constant and is equal to the length of transverse axis of the hyperbola.


A second definition of the hyperbola

On account of this property, a second definition of the hyperbola is:

A hyperbola is the locus of a point which moves in such a way that the difference of its distance from two fixed points (foci) is always constant.

Equation of Hyperbola in standard form

x²/a² - y²/b² = 1

Second Focus and Second Directrix of the hyperbola

Hyperbola with equation x²/a² - y²/b² = 1
has a second focus at S’ = (-ae,0) and second directrix (Z’K’) with equation x = -a/e

Various Measures of Hyperbola

Foci,Vertices, Transverse and Conjugate Axes, Directrices and Centre of the Hyperbola x²/a² - y²/b² = 1

Foci: The points S(ae,0) and S’ (-ae,0) are the foci of the hyperbola.

Vertices: Where curve meets the line joining the foci S and S’, are called the vertices of the hyperbola (A,A’).

Transverse Axis: The straight line joining the vertices A and A’ is called the transverse axis of the hyperbola. Its length is taken to be 2a.

Centre of the hyperbola:

The middle point of AA’ is called the centre of the hyperbola. It bisects every chord of the hyperbola that passes through it.

Conjugate axis:

The straight line through the centre which is perpendicular to the transverse axis does not meet the hyperbola in real points.

But if B, B’ are points on this line such that CB = CB’ = b, the line BB’ is called the conjugate axis such that BB’ = 2b.

Eccentricity of the hyperbola

Eccentricity of the hyperbola x²/a² - y²/b² = 1

e = √(1+b²/a²)

e = √(1+(conjugate axis)²/(transverse axis)²)

Length of the latus rectum of the hyperbola

Length of the latus rectum of the hyperbola x²/a² - y²/b² = 1

is 2b²/a = 2a(e²-1)

Focal distances of a point

The difference of the focal distances of any point on a hyperbola is constant and equal to the length of the transverse axis of the hyperbola.

Conjugate Hyperbola

The hyperbola whose transverse and conjugate axes are respectively the conjugate and transverse axes of a given hyperbola is called the conjugate hyperbola of the given hyperbola.

The conjugate hyperbola of the Hyperbola with Equation x²/a² - y²/b² = 1 is -x²/a² + y²/b² = 1.

Parametric Coordinates and parametric equations of hyperbola

x = a sec θ and y = b tan θ satisfy equation x²/a² - y²/b² = 1 as the transformation turns out to be sec² θ - tan² θ which is equal to 1.

Hence the parametric coordinates of hyperbola are (a sec θ,b tan θ) and x = a sec θ and y = b tan θ are called parametric equations of hyperbola.

x = a cosh θ and y = a sinh θ are also parametric equations of hyperbola.

cosh θ = (e^θ+e^-θ)/2


sinh θ = (e^θ-e^-θ)/2

Equation of the chord joining any two points on a hyperbola

Hyperbola
Equation x²/a² - y²/b² = 1

Two points are P(a sec θ1, b tan θ1) and Q(a sec θ2, b tan θ2)

Equation of the chord joining two points P and Q is

y - b tan θ1 = (x - a sec θ1)[b tan θ2 - b tan θ1]/[a sec θ2-a sec θ1]

Intersection of a line and a hyperbola

Line y = mx+c

Hyperbola

x²/a² – y²/b² = 1


The equation which gives intersection points

x²(a²m²-b²) + 2a²mcx + a²(c²+b²) = 0

The two values of x and y may be real and distinct, coincident or imaginary.

Condition of a line to be a tangent to a hyperbola

The line y = mx+c is a tangent to the hyperbola if c² = a²m²-b²

Equation of tangent to a hyperbola in different forms

Hyperbola

x²/a² – y²/b² = 1

point (x1,y1)



Slope form

The line y = mx+c is a tangent to the hyperbola if c² = a²m²-b² .
Hence, the line y = mx±SQRT(a²m²-b²) is always a tangent.
The points of contact are (±a²m/c,±b²/c)

Point form

Tangent at (x1,y1) is

xx1/a²-yy1/b² = 1

Parametric form

point (a sec θ, b tan θ)

(x sec θ)/a) - ((y tan θ)/b) = 1

Number of tangents drawn from a point to a hyperbola

Two tangents can be drawn from a point to a hyperbola

Equation of the pair of tangents from a point to a hyperbola

Hyperbola

x²/a² – y²/b² = 1

point (x1,y1)


[x²/a² – y²/b² - 1][x1²/a² – y1²/b² - 1] = [xx1/a² – yy1/b² - 1]²

can be written as SS1 = T²

Equation of the chord of contacts of tangents for hyperbola

Hyperbola

x²/a² – y²/b² = 1

point (x1,y1)


chord of contacts is xx1/a² - yy1/b² = 1

Equation of normal to hyperbola in different forms

Point form: Normal at (x1,y1)to hyperbola x²/a² - y²/b² = 1 is

x-x1/(x1/a²) = -(y-y1)/(y1/b²)

The equation can also be expressed as

a²x/x1 + b²y/y1 = a² + b²

Parametric form: Normal at (a sec θ , b tan θ)to hyperbola x²/a² - y²/b² = 1 is


a x cos θ + by cot θ = a² + b²

Number of normals

In general four normals can be drawn from a point (x1,y1) to the hyperbola.

Asymptotes of a hyperbola

An asymptote to a curve is a straight line, at a finite distance from the origin, to which the tangent to a curve tends as the point of contact goes to infinity. In other words, asymptote to a curve touches the curve at infinity.

Rectangular hyperbola

Defintion: a hyperboal whose asymptotes are at right angles to each other is called a rectangular hyperbola.

Equation is x² - y² = a²

Evaluation of Limits by Using DE 'L' Hospital's Rule

If f(x0 and g(x) are two functions of x and

(i) lim (x→a) f(x) = lim (x→a) g(x) = 0

both functions are continuous and differentiable at x = a

Then lim (x→a) f(x)/g(x) = lim (x→a) f'(x)/g'(x)

Provided g(a)≠ 0 and f'(x) and g'(x) are continuous at x=a.

Intervals (Closed and open)

Closed interval

The set of all real numbers x such that a≤x≤b ( a and b are also real numbers) is called a closed interval and is denoted by [a,b].

Open interval

The set of all real numbers x such that a is l.t.x is l.t. b ( a and b are also real numbers) is called a one interval and is denoted by (a,b).

Domains and ranges of real functions

Domain of a real function is the set of all real numbers for which f(x) is real number.

Real functions - Examples

Constant function
Identify function
Modulus
The greatest integer
Reciprocal
Logarithmic
Square root
Polynomial
Rational
Trigonometrical
Inverse trigonometrical

Operations on real functions

Sum
Difference
Product
Quotient
Scalar multiple
Composition of functions

Extension of a function

Extending a function from a given interval [0,a] to [-a,a].

Periodic function

A function f(x) is said to be a periodic function if there exists a positive real number T such that f(x+T) = f(x) for all x Є R.



Periodic functions and their periods

sin x 2 π

cos x 2 π

tan x π


Some Results on periodic functions

If f(x) is a periodic function with period T and a,b Є R such that a is g.t. 0, then f(ax+b) is periodic with period T/a.


If (f1(x), f2(x) and f3(x) are periodic functions with periods T1,T2,and T3 respectively, then a1f1(x)+a2f2(x)+a3f3(x) is a periodic function with period equal to LCM of T1,T2,T3 and where a1,a2, and a3 are non-zero real numbers.

Formal approach to limit

A real number l is limit of f(x) as x tends to a, if for every ε>0 there exists a δ>0 such that

0<|x-a|< δ => |f(x)-l|< ε
<=> x.( a- δ, a+δ), x≠a => f(x).(l- ε, l+ ε)

| f(x)-l| < ε means f(x) belong to the ε neighbourhood of l and | x-a| < δ (and x≠a) means x belongs to the deleted δ neighbourhood of a.

The open interval (a- δ, a+ δ) whose length is 2 δ and whose midpoint is a, is called the δ-interval of a. Open interval means a- δ, and a+ δ are not part of the interval.

{x/x Є(a- δ,a+ δ), x≠a} is called the deleted δ-neighbourhood of a.

Evaluation of left hand and right hand limits

The statement x→aˉ means that x is tending to a from the left hand side.

The statement x→a⁺ means that x is tending to a from the right hand side.

Steps to find left hand limit

Put x = a-h and replace x→aˉ by h→0. Find limit of f(a-h) as h→0


Steps to find right hand limit

Put x = a+h and replace x→a⁺ by h→0. Find limit of f(a+h) as h→0

Some Standard limits

1. lim x→a x = a

2. lim x→a kx = ka, k Є R

3. lim x→a x^k = a^k , k Є R

4. lim x→a k = k


5. lim x→a (x^k –a^k )/(x-a) = na^n-1, nk Є N

6. lim x→a sin x = sin α

7. lim x→a cos x = cos α

8. lim x→a (sin x)/x = 1, where x is measured in radians

The algebra of limits

1. lim x→a [f(x) ±g(x)] = lim x→a f(x) ±lim x→ag(x)

2. lim x→a [k.f(x)] = k lim x→af(x)

3. lim x→a[f(x).g(x)] = [lim x→a f(x)][ lim x→a g(x)]

4. lim x→a [f(x)/g(x)] = [lim x→a f(x)]/[ lim x→a g(x)]
provided lim x→a g(x) ≠ 0.



5. If f(x) is l.t. g(x) then lim x→a f(x) ≤lim x→a g(x)

Evaluation of limits

Evaluation of Algebraic limits

Direct substitution method
Factorisation method
Rationalization method
Using standard formulas
Method for limits when x→∞

Evaluation of Trigonometric limits

Evaluation of Exponential and Logarithmic Limits

Evaluation of Exponential Limits of the Form 1ˉ

Continuity at a point

a function f(x) is said to be continuous at a point x = a of its domain iff lim (x→a) = f(a)

Related concepts

Discontinuity

Removable Discontinuity

Discontinuity of first kind

Discontinuity of second kind

Continuity of functions in an interval

A function f(x) is said to be continuous on an open interval (a,b) iff it is continuous at every point on the interval (a,b).

A function f(x) is said to be continuous on a closed interval [a,b] iff

f is continuous at every point on the interval (a,b), i.e., f is continuous on the open interval (a,b) and

lim (x→a⁺) f(x) = f(a) and lim (x→bˉ) = f(b).

It has to be continuous on (a,b), it has to be continuous at a from right and at b from left.

Continuous functions

A function f(x) is said to be continuous, if it is continuous at each point of its domain.

Everywhere continous function:

A function f(x) is said to be everywhere continuous, if it is continuous on the entire real line (-∞,∞).

Cauchy’s definition of continuity

a function f is said to be continuous at a point a of its domain D iff for every ε>0 there exists a δ>0 (dependent on ε) such that

|x-a| <δ => |f(x)-f(a)| < ε

Heine’s definition of continuity

A function f is said to be continuous at a point a of its domain D, if for every sequence a_n of the points in D converging to a, the sequence of f(a_n) converges to f(a)

i.e., lim a_n = a => lim f(a_n) = f(a).

21.9 Differentiability at a point

A function f(x) is said to be differntiable or derivable at x+c,
iff lim (x→c) [f(x)-f(c)/(x-c)] exists finitely.

Relation between continuity and differentiability

If a function is differentiable at a point, it is necessarily continuous at that point.

But the converse is not necessarily true.

This means a function may be continuous at a point but may not be differentiable at that point.

Differentiability in a set

A function f(x) defined on an openinterval (a,b) is said to be differentiable in open interval (a,b) if it is differentiable at each point of (a,b).

Some results on differentiability

Every polynomial function is differentiable at each x Є R.

The exponential function is differentiable a^x, a>o is differentiable at each x Є R.

Every constant function is differentiable at each x Є R.

The logarithmic function is differentiable at each point in its domain.

The trigonometric and inverse-trigonometrics functions are differentiable at each point in their domains.

The sum, difference, product and quotient of two differentiable functions are differentiable.

The composition of differentiable function is a differential function.