I studied the chapters limits and continuity in XI and XII books and R D Sharma also during the last two days. I studied the chapter on Binomial theorem also.
19 June 2008
Studied the chapter on differential equations.
21 June 2008
Studied the chapters on Straight line and Family of lines and posted the revision points.
Sunday, June 15, 2008
Saturday, June 7, 2008
IIT JEE Revision Points
I prepared the structure for writing down revision points for each chapter of Mathematics neatly. I posted some points in some chapters. I shall complete all the chapters over a period of time. I feel this material will be very useful once we get into revision mode. Then time is very important and any structure that allows a very quick revision will be very very helpful. My feeling is that revision point structure will allow full revision of the subject in a day. As I am developing similar structure for chemistry (I completed it to a large extent)and physics, a three day revision of very important points of the entire syllabus is possible. That will give very good advantage from January 1, 2009, as from that day onwards, I hope we will be in full revision mode.
25-6-2008
Updated the parabola revision points
25-6-2008
Updated the parabola revision points
Ch. 1. Sets - Number of elements in sets
Some results on Number of elements in sets n(A), n(B), and n(C)
Note union operation and universal set have the same symbol
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)
Laws of algebra of sets
1. Idempotent laws
A U B = A
A∩A = A
2. Identity laws
3. Commutative laws
4. Associative laws
5. Distributive laws
6. De-Morgan's laws
Note union operation and universal set have the same symbol
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)
Laws of algebra of sets
1. Idempotent laws
A U B = A
A∩A = A
2. Identity laws
3. Commutative laws
4. Associative laws
5. Distributive laws
6. De-Morgan's laws
Ch. 3. Functions - Concept Review
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.
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 nm.
Types of functions
1. Even and odd functions
2. Monotonic functions
3. Step function
4. Modulus function
5. Algebraic function
6.Exponential function
7. Logarithmic function
8. Inverse function
9. composite function
10. Trigonometric function
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.
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 nm.
Types of functions
1. Even and odd functions
2. Monotonic functions
3. Step function
4. Modulus function
5. Algebraic function
6.Exponential function
7. Logarithmic function
8. Inverse function
9. composite function
10. Trigonometric function
Ch. 3. Functions - 2 (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.
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.
Ch. 4. Binary Operations - 1
Binary operation
Let S be a non-void set. A function from S×S to S is called a binary operation on S.
f:S×S S is binary operation on set S
A binary function f on a set S×S associates each order pair (a,b) of elements of S×S to a unique element f(a,b) of S.
Instead of writing f(a,b) for the image of an ordered pair we can write a f b.
a f b = f(a,b)
Binary operations are represented by the symbols *, etc. instead of letter f,g etc.
Instead S×S only Set S is mentioned in binary operation but it is to be understood as S×S.
Thus a binary operation * on a set S associates each ordered pair (a,b) of elements of S to a unique element a*b of S. (a*b is an element in S that is obtained by the relation a*b -- * represents relation on set S and set S)
Example: Addition of two natural numbers.
(a,b) is an element of N×N (a and b are natural numbers)
a+b is also a natural number and is also an element of N.
Addition of natural numbers is f:N×NN a binary operation.
Types of binary operations
Commutative binary operation.
Associative binary operation.
Distributive binary operation
Identity and inverse elements
Identity element
Inverse of an element
Composition table
A binary operation on finite set can be completely described by means of a table known as a composition table.
Let S be a non-void set. A function from S×S to S is called a binary operation on S.
f:S×S S is binary operation on set S
A binary function f on a set S×S associates each order pair (a,b) of elements of S×S to a unique element f(a,b) of S.
Instead of writing f(a,b) for the image of an ordered pair we can write a f b.
a f b = f(a,b)
Binary operations are represented by the symbols *, etc. instead of letter f,g etc.
Instead S×S only Set S is mentioned in binary operation but it is to be understood as S×S.
Thus a binary operation * on a set S associates each ordered pair (a,b) of elements of S to a unique element a*b of S. (a*b is an element in S that is obtained by the relation a*b -- * represents relation on set S and set S)
Example: Addition of two natural numbers.
(a,b) is an element of N×N (a and b are natural numbers)
a+b is also a natural number and is also an element of N.
Addition of natural numbers is f:N×NN a binary operation.
Types of binary operations
Commutative binary operation.
Associative binary operation.
Distributive binary operation
Identity and inverse elements
Identity element
Inverse of an element
Composition table
A binary operation on finite set can be completely described by means of a table known as a composition table.
Ch. 5. Complex Numbers - 1
1. Introduction
Sqrt(-1) = i
“i” is called imaginary unity
2. Integral powers of IOTA (i)
i³ = i*i² = i*(-1) = -i
To find in divide n by 4 to get 4m+r where m is the quotient and r is the remainder.
i4 = 1
in will be equal to ir
3. Imaginary quantities
Square root of -3, -5 etc are called imaginary quantities
4. complex numbers
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)).
5. Equality of complex numbers
6. Addition of complex numbers
7. Subtraction of complex numbers
8. 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))
9. Division of complex numbers
z1/z2 = z1* Multiplicative inverse of z2
10. Conjugate of a complex number
conjugate of z (= a+ib) = a-ib (is termed as z bar)
11. 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
12, Reciprocal of a complex number
Multiplicative inverse and reciprocal are same
13. Square root of a complex number
14. Representation of a complex number
Graphical – Argand plane
Trigonometric
Vector
Euler
15. Argument or amplitude of a complex number
16. Eulerian form of a complex number
eθ = cosθ + i sinθ and e-θ = cos θ - i sin θ
These two are called Eulerian forms of a complex number.Sqrt(-1) = i
“i” is called imaginary unity
2. Integral powers of IOTA (i)
i³ = i*i² = i*(-1) = -i
To find in divide n by 4 to get 4m+r where m is the quotient and r is the remainder.
i4 = 1
in will be equal to ir
3. Imaginary quantities
Square root of -3, -5 etc are called imaginary quantities
4. complex numbers
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)).
5. Equality of complex numbers
6. Addition of complex numbers
7. Subtraction of complex numbers
8. 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))
9. Division of complex numbers
z1/z2 = z1* Multiplicative inverse of z2
10. Conjugate of a complex number
conjugate of z (= a+ib) = a-ib (is termed as z bar)
11. 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
12, Reciprocal of a complex number
Multiplicative inverse and reciprocal are same
13. Square root of a complex number
14. Representation of a complex number
Graphical – Argand plane
Trigonometric
Vector
Euler
15. Argument or amplitude of a complex number
16. Eulerian form of a complex number
eθ = cosθ + i sinθ and e-θ = cos θ - i sin θ
17. Geometrical representations of fundamental operations
Addition
Subtraction
17a. Modulus and argument of multiplication of two complex numbers
18. Modulus and argument of division of two complex numbers
19. Geometrical representation of conjugate of a complex number
20. Some important results on modulus and argument
Ch. 6. Sequences and Series - 1
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.
2 Arithmetic progression
3. General term of A.P.
nth term = a +(n-1)d
6.4 Selection of terms in A P.
6.5 Sum to n terms of AP
Sn = ½ n(a + l)
Where
n = number of terms in the progressin
a = first term
l = last term
Sn = ½ n{2a+(n-1)d}
6. Properties of AP
7. Insertion of arithmetic means
When three quantities are in arithmetical progression, the middle one is called the arithmetic mean of the other two.
a-d, a, and a+d are in arithmetic progression. ‘a’ is the arithmetic mean of a-d and a+d.
For model problems on A.P. visit
http://iit-jee-maths-aps.blogspot.com/2008/06/progressions-model-problems-1.html
8. Geometric progression (GP)
9. General term of a GP
nth term = arn-1
10. Selection of terms in a GP
11. sum of n terms of GP
Sn = a(1-rn)/(1-r)
12. Sum of infinite GP
S = a/(1-r) when -1
13. Properties of GP
14. Insertion of Geometric mean between two given numbers
When three quantities are in geometrical progression, the middle one is said to be the geometric mean of the other two.
a/r, a, and ar are in GP. ‘a’ is the geometric mean of a/r and ar.
√(ab) is the geometric mean of a and b
15. Some important properties of AM and GM between two quantities
16. Arithmetic Geometric sequence
17. Sum of n terms of arithmetic geometric sequence
18. Sum to n terms of some special sequences
19. Miscellaneous sequences and series
20. harmonic progressions
21. Properties of arithmetic, geometric and harmonic means between two given numbers.
For model problems on this chapter visit
http://iit-jee-maths-aps.blogspot.com/2008/06/progressions-model-problems-1.html
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.
2 Arithmetic progression
3. General term of A.P.
nth term = a +(n-1)d
6.4 Selection of terms in A P.
6.5 Sum to n terms of AP
Sn = ½ n(a + l)
Where
n = number of terms in the progressin
a = first term
l = last term
Sn = ½ n{2a+(n-1)d}
6. Properties of AP
7. Insertion of arithmetic means
When three quantities are in arithmetical progression, the middle one is called the arithmetic mean of the other two.
a-d, a, and a+d are in arithmetic progression. ‘a’ is the arithmetic mean of a-d and a+d.
For model problems on A.P. visit
http://iit-jee-maths-aps.blogspot.com/2008/06/progressions-model-problems-1.html
8. Geometric progression (GP)
9. General term of a GP
nth term = arn-1
10. Selection of terms in a GP
11. sum of n terms of GP
Sn = a(1-rn)/(1-r)
12. Sum of infinite GP
S = a/(1-r) when -1
13. Properties of GP
14. Insertion of Geometric mean between two given numbers
When three quantities are in geometrical progression, the middle one is said to be the geometric mean of the other two.
a/r, a, and ar are in GP. ‘a’ is the geometric mean of a/r and ar.
√(ab) is the geometric mean of a and b
15. Some important properties of AM and GM between two quantities
16. Arithmetic Geometric sequence
17. Sum of n terms of arithmetic geometric sequence
18. Sum to n terms of some special sequences
19. Miscellaneous sequences and series
20. harmonic progressions
21. Properties of arithmetic, geometric and harmonic means between two given numbers.
For model problems on this chapter visit
http://iit-jee-maths-aps.blogspot.com/2008/06/progressions-model-problems-1.html
Ch. 6. Sequences and Series - 2
Arithmetico Geometric sequence
Arithmetico-geometric sequence
When a1, a2, a3,…an is an A.P. and b1,b2,b3,…,bn are in GP
a1b1, a2b2, a3b3 Are said to be in arithmetico-geometric sequence.
In terms common difference and common ratio, the series can be written as
ab, (a+d)br, (a+2d)br²
nth term = [a+(n-1)d]brn-1
Sum to n term of an A-G series
Sn = ab+ (a+d)br+ (a+2d)br²+…+[a+(n-2)d]brn-2 + [a+(n-1)d]brn-1
Multiply by each of Sn by r
r*Sn = abr + (a+d)br²+ (a+2d)br³+…+[a+(n-2)d]brn-1
+ [a+(n-1)d]brn
Sn – r*Sn = ab+dbr+d br² +d br³ +…+d brn-1 - [a+(n-1)d]brn
=> ab+dbr[1+r +r² +r³ +…+rn-2] - [a+(n-1)d]brn
As [1+r +r² +r³ +…+rn-2] = (1-rn-1)/(1-r) (as there are n-1 terms in the GP)
(1-r)Sn = ab + dbr(1-rn-1)/(1-r) - [a+(n-1)d]brn
Sn = ab/(1-r) + dbr(1-rn-1)/(1-r)² - [a+(n-1)d]brn/(1-r)
Sum to n terms of an A-G series =
Sn = ab/(1-r) + dbr(1-rn-1)/(1-r)² - [a+(n-1)d]brn/(1-r)
Arithmetico-geometric sequence
When a1, a2, a3,…an is an A.P. and b1,b2,b3,…,bn are in GP
a1b1, a2b2, a3b3 Are said to be in arithmetico-geometric sequence.
In terms common difference and common ratio, the series can be written as
ab, (a+d)br, (a+2d)br²
nth term = [a+(n-1)d]brn-1
Sum to n term of an A-G series
Sn = ab+ (a+d)br+ (a+2d)br²+…+[a+(n-2)d]brn-2 + [a+(n-1)d]brn-1
Multiply by each of Sn by r
r*Sn = abr + (a+d)br²+ (a+2d)br³+…+[a+(n-2)d]brn-1
+ [a+(n-1)d]brn
Sn – r*Sn = ab+dbr+d br² +d br³ +…+d brn-1 - [a+(n-1)d]brn
=> ab+dbr[1+r +r² +r³ +…+rn-2] - [a+(n-1)d]brn
As [1+r +r² +r³ +…+rn-2] = (1-rn-1)/(1-r) (as there are n-1 terms in the GP)
(1-r)Sn = ab + dbr(1-rn-1)/(1-r) - [a+(n-1)d]brn
Sn = ab/(1-r) + dbr(1-rn-1)/(1-r)² - [a+(n-1)d]brn/(1-r)
Sum to n terms of an A-G series =
Sn = ab/(1-r) + dbr(1-rn-1)/(1-r)² - [a+(n-1)d]brn/(1-r)
Ch. 6. Sequences and Series - 4 - Harmonic Progression
Harmonic progression - Revision points
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 mean between two numbers a, b
A = (a+b)/2
G = √ab
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.
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 mean between two numbers a, b
A = (a+b)/2
G = √ab
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.
Ch. 9 Binomial Theorem - 1
If x and a are real numbers, then for all n Є N
(x+a)n
= nC0xna0 + nC1xn-1a1 +nC2xn-2a2 + ...+nCrxn-rar+ ...+nCn-1x1an-1+nCnx0an
(x+a)n
= (r = 0 to n)Σ nCrxn-rar
Some conclusions
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*nCrxn-rar
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 nCrxr
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 nCrxn-r
7. (1-x) n = (r = 0 to n)Σ(-1)r* nCrxr
8. (x+a) n +(x-a) n = 2[nC0xna0 +
nC2xn-2a2 +
nC4xn-4a4+ ...]
9. General term in a binomial expansion
(r+1) term in (x+a) n
= nCrxn-rar
10. Another form of binomial expansion
(x+a)n
= (r = 0 to n) and r+s = n Σ (n!/r!s!) xras
11. Coefficient of (r+1)th term in the binomial expression of (1+x)n is nCr
12. Algorithm to find the greatest term
(i) Write term r+1 = T(r+1) and term r = T® from the given expression.
(ii) Find T(r+1)/T®
(iii) Put T(r+1)/T®>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
13 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 2n-1.
14. Middle terms in binomial expression
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
Multinomial theorem
(x1+x2)n = (r1 = 0 to n) and r1+r2 = nΣ (n!/r1!r2!) x1r1x2r2
(x+a)n
= nC0xna0 + nC1xn-1a1 +nC2xn-2a2 + ...+nCrxn-rar+ ...+nCn-1x1an-1+nCnx0an
(x+a)n
= (r = 0 to n)Σ nCrxn-rar
Some conclusions
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*nCrxn-rar
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 nCrxr
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 nCrxn-r
7. (1-x) n = (r = 0 to n)Σ(-1)r* nCrxr
8. (x+a) n +(x-a) n = 2[nC0xna0 +
nC2xn-2a2 +
nC4xn-4a4+ ...]
9. General term in a binomial expansion
(r+1) term in (x+a) n
= nCrxn-rar
10. Another form of binomial expansion
(x+a)n
= (r = 0 to n) and r+s = n Σ (n!/r!s!) xras
11. Coefficient of (r+1)th term in the binomial expression of (1+x)n is nCr
12. Algorithm to find the greatest term
(i) Write term r+1 = T(r+1) and term r = T® from the given expression.
(ii) Find T(r+1)/T®
(iii) Put T(r+1)/T®>1
(iv) Solve the inequality for r to get an inequality of the form r
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
13 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 2n-1.
14. Middle terms in binomial expression
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
Multinomial theorem
(x1+x2)n = (r1 = 0 to n) and r1+r2 = nΣ (n!/r1!r2!) x1r1x2r2
Logarithms - Basic points
Let a,b be two positive real numbers and a≠1. The real number x such that ax = b is called logarithm of b to the base a.
X = logab
Theorems
1. If a is a positive number and a≠1 then logaa = 1.
2. If a is a positive number and a≠1 then loga1 = 0
3. If a,m are positive real numbers, a≠1 then a to the power logam = m.
4. If a,m,n are positive real numbers and a≠1 then logamn = logam + logan
5. If a,m,n are positive real numbers and a≠1 then logam/n = logam - logan
6. If a,m,n are positive real numbers and a≠1 then logamn = nlogam
7. If a,b,n are positive real numbers and a≠1, b≠1 then logam = logbm* logab
8. If a>1, then x>y => logax > logay
9. If 0y => logax < logay
e = 1+1/1! +1/2! + 1/3! + ¼! + …∞
e = lim n →∞ (1+(1/n)) n
If an = x, the logax = n.
Log (1=x) when |x|<1 = x-x²/2 +x³/3 - x4/4+…∞
X = logab
Theorems
1. If a is a positive number and a≠1 then logaa = 1.
2. If a is a positive number and a≠1 then loga1 = 0
3. If a,m are positive real numbers, a≠1 then a to the power logam = m.
4. If a,m,n are positive real numbers and a≠1 then logamn = logam + logan
5. If a,m,n are positive real numbers and a≠1 then logam/n = logam - logan
6. If a,m,n are positive real numbers and a≠1 then logamn = nlogam
7. If a,b,n are positive real numbers and a≠1, b≠1 then logam = logbm* logab
8. If a>1, then x>y => logax > logay
9. If 0
e = 1+1/1! +1/2! + 1/3! + ¼! + …∞
e = lim n →∞ (1+(1/n)) n
If an = x, the logax = n.
Log (1=x) when |x|<1 = x-x²/2 +x³/3 - x4/4+…∞
Ch. 11 Matrices - 1
Matrix is asset 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 matrx’.
Types of matrices
Row matrix
Column matrix
Diagonal matrix
Scalar matrix
Identity or unit matrix
Upper triangular matrix
Lower triangular matrix
Types of matrices
Row matrix
Column matrix
Diagonal matrix
Scalar matrix
Identity or unit matrix
Upper triangular matrix
Lower triangular matrix
Ch. 11 Matrices - Inverse - Concept Review 3
Inverse of a matrix
Let A be a square matrix of order n
If AB = In = 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 AT is also invertible and
(AT)-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 AT = (adj A) T
11. If A is a non-singular square matrix, then
adj(adj A) = |A|n-2A
Let A be a square matrix of order n
If AB = In = 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 AT is also invertible and
(AT)-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 AT = (adj A) T
11. If A is a non-singular square matrix, then
adj(adj A) = |A|n-2A
Ch. 12 Determinants - 2
12.9
Determinants coordinate geometry
The area of a triangle having vertices at (x1,y1), (x2,y2) an d(x3,y3) us given by
Δ = determinants of
|x1 y1 1|
|x2 y2 1| *(1/2)
|x3 y3 1|
Condition of collinearity of three points:
Let the three points be (x1,y1), (x2,y2) and (x3,y3). The area of triange formed by these three points is zero. Hence the determinant
|x1 y1 1|
|x2 y2 1| will be zero
|x3 y3 1|
Equation of a line passing through two points:
Let the two points be (x1,y1), (x2,y2). If (x,y) is a point on the line passing through these two points, then (x1,y1), (x2,y2) and (x,y) are collinear. Hence the determinant
|x y 1|
|x1 y1 1|
|x2 y2 1| will be zero
Determinants coordinate geometry
The area of a triangle having vertices at (x1,y1), (x2,y2) an d(x3,y3) us given by
Δ = determinants of
|x1 y1 1|
|x2 y2 1| *(1/2)
|x3 y3 1|
Condition of collinearity of three points:
Let the three points be (x1,y1), (x2,y2) and (x3,y3). The area of triange formed by these three points is zero. Hence the determinant
|x1 y1 1|
|x2 y2 1| will be zero
|x3 y3 1|
Equation of a line passing through two points:
Let the two points be (x1,y1), (x2,y2). If (x,y) is a point on the line passing through these two points, then (x1,y1), (x2,y2) and (x,y) are collinear. Hence the determinant
|x y 1|
|x1 y1 1|
|x2 y2 1| will be zero
Ch.13 Straight Lines - Revision Points 1
1. The equation of a line parallel to x-axis is of the form y = k.
The equation of a line parallel to y axis is of the form x = k, where k is a constant.
2. If a line makes an angle θ with the positive direction of x-axis and θ ≠π/2, then the slope of the line is given by tan θ.
3. the slope of a line passing through (x1,y1) and (x2,y2) is (y2-y1)/(x2-x1), provided x1≠x2.
4. If two lines have finite slopes m1 and m2
then they are parallel iff m1 = m2
they are perpendicular iff m1*m2 = -1
5. The equation of a line having slope m and y intercept c is y = mx+c
6. The equation of a line having slope m and passing through (x1,y1) is
(y-y1) = m(x-x1)
7. The equation of a line having slope m and passing through (x1,y1)and (x2,y2) is
(y-y1)/(x-x1) = (y1-y2)/(x1-x2)
8.The equation of a line making non-zero intercepts and b on the x and y axes respectively is
(x/a) + (y/b) = 1
9. The equation of a line such that the perpendicular drawn from the origin to the line has length p and inclination α is
x cos α + y sin α = p.
10. The general equation of a line is of the form ax +by +c = o and its slope is –a/b, provided b≠0.
11. 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.
12. The perpendicular distance of (x1,y1) from the line ax+by+c = 0 is given by |ax1 + by1 + c|/| √(a² + b²)|
13. The point of intersection of two lines, which are not parallel, can be found by solving their equations simultaneously.
14. Family of lines
If u ≡ a1x + b1y +c1 = 0 and
v≡ a2x +b2y +c2 = 0
the u + kv = 0, k Є R represents a family of lines
(i) if u and v are intersecting lines, then u + kv = 0, k Є R represents a family of lines passing through the point of intersection of u =0 and v=0.
(ii) if u and v are two parallel lines, u + kv = 0, k Є R represents a family of straight lines parallel to u =0 and v=0.
The equation of a line parallel to y axis is of the form x = k, where k is a constant.
2. If a line makes an angle θ with the positive direction of x-axis and θ ≠π/2, then the slope of the line is given by tan θ.
3. the slope of a line passing through (x1,y1) and (x2,y2) is (y2-y1)/(x2-x1), provided x1≠x2.
4. If two lines have finite slopes m1 and m2
then they are parallel iff m1 = m2
they are perpendicular iff m1*m2 = -1
5. The equation of a line having slope m and y intercept c is y = mx+c
6. The equation of a line having slope m and passing through (x1,y1) is
(y-y1) = m(x-x1)
7. The equation of a line having slope m and passing through (x1,y1)and (x2,y2) is
(y-y1)/(x-x1) = (y1-y2)/(x1-x2)
8.The equation of a line making non-zero intercepts and b on the x and y axes respectively is
(x/a) + (y/b) = 1
9. The equation of a line such that the perpendicular drawn from the origin to the line has length p and inclination α is
x cos α + y sin α = p.
10. The general equation of a line is of the form ax +by +c = o and its slope is –a/b, provided b≠0.
11. 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.
12. The perpendicular distance of (x1,y1) from the line ax+by+c = 0 is given by |ax1 + by1 + c|/| √(a² + b²)|
13. The point of intersection of two lines, which are not parallel, can be found by solving their equations simultaneously.
14. Family of lines
If u ≡ a1x + b1y +c1 = 0 and
v≡ a2x +b2y +c2 = 0
the u + kv = 0, k Є R represents a family of lines
(i) if u and v are intersecting lines, then u + kv = 0, k Є R represents a family of lines passing through the point of intersection of u =0 and v=0.
(ii) if u and v are two parallel lines, u + kv = 0, k Є R represents a family of straight lines parallel to u =0 and v=0.
Thursday, June 5, 2008
Ch. 14 Family of Lines - 1
The joint equation of the straight lines a1x +b1y +c1 = 0 and a2x +b2y+ c2 = 0 is
(a1x +b1y +c1) (a2x +b2y+ c2) = 0
ax² +2hxy+by²+2gx+2fy+c = 0 is joint equation of 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.
Bisectors of the angle between the lines given by a homogeneous equation
The jont equation of the bisectors of the angles between the lines represented by ax² +2hxy+by² = 0 is given by
(x²-y²)/(a-b) = xy/h
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.
Algorithm to find the joint equation of 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.
(a1x +b1y +c1) (a2x +b2y+ c2) = 0
ax² +2hxy+by²+2gx+2fy+c = 0 is joint equation of 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.
Bisectors of the angle between the lines given by a homogeneous equation
The jont equation of the bisectors of the angles between the lines represented by ax² +2hxy+by² = 0 is given by
(x²-y²)/(a-b) = xy/h
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.
Algorithm to find the joint equation of 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.
Ch. 15 The Circle - Formula Review
1. 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
2. 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
3. General equation of a circle
x²+y²+2gx+2fy+c = 0
Centre of this circle = (-g,-f)
Radius = √(g²+f²-c)
4. 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
5. 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)
6. 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. Let the radius of the circle be R
If CP
7. 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
8. 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
9. 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) ]
10.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
11 Normal to a circle at a given point
If slope of the tangent is m, then the slope of the normal is –(1/m)
12. 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)
13A. Pair of tangents drawn from a point to given circle
Let the point be (x1,y1) and the circle be x² + y² = a²
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
13B. Combined equation of pair of tangents
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
14. Director circle and its equation
Equation of director circle of the circle x² + y² = a² is x² + y² = 2a²
15. Chord of contacts of tangents
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.
16. Pole and Polar
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.
17. 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
18. 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
19. 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.
20. 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.
21. 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
22. Radical axis
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.
23. 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.
24. 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
25. 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.
(x-h)²+(y-k)² = a²
Centre of the circle is at (h,k)
radius of the circule is a
2. 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
3. General equation of a circle
x²+y²+2gx+2fy+c = 0
Centre of this circle = (-g,-f)
Radius = √(g²+f²-c)
4. 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
5. 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)
6. 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. Let the radius of the circle be R
If CP
7. 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
8. 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
9. 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) ]
10.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
11 Normal to a circle at a given point
If slope of the tangent is m, then the slope of the normal is –(1/m)
12. 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)
13A. Pair of tangents drawn from a point to given circle
Let the point be (x1,y1) and the circle be x² + y² = a²
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
13B. Combined equation of pair of tangents
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
14. Director circle and its equation
Equation of director circle of the circle x² + y² = a² is x² + y² = 2a²
15. Chord of contacts of tangents
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.
16. Pole and Polar
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.
17. 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
18. 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
19. 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.
20. 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.
21. 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
22. Radical axis
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.
23. 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.
24. 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
25. 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.
Ch. 16 Parabola - Concept Review
Revision Points
1. Conic sections: Definition
2. The parabola
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 iine 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
3. 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|
4. 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
5. Position of a point with respect to a parabola
6. Equation of a parabola in parametric form
x = at²
y = 2at
It satisfies y² = 4ax
y² = 4a²t²
4ax = 4a²t²
7. 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
8. 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
9. Equation of tangent in different forms
Parabola equation y² = 4ax; 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
10. 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
11 Number of normals drawn from a point to a parabola
In general three normals can be drawn from a point to a parabola
12. Some results in conormal points
The sumof the slopes of the normals at conormal points is zero.
13 Number of tangents drawn from a point to a parabola
Two tangents can be drawn from a point to a parabola
13a. 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)
14. 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)
15. 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
16. 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.
17. Length 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)
18. Pole and Polar
19. some important results at a glance
1. Conic sections: Definition
2. The parabola
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 iine 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
3. 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|
4. 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
5. Position of a point with respect to a parabola
6. Equation of a parabola in parametric form
x = at²
y = 2at
It satisfies y² = 4ax
y² = 4a²t²
4ax = 4a²t²
7. 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
8. 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
9. Equation of tangent in different forms
Parabola equation y² = 4ax; 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
10. 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
11 Number of normals drawn from a point to a parabola
In general three normals can be drawn from a point to a parabola
12. Some results in conormal points
The sumof the slopes of the normals at conormal points is zero.
13 Number of tangents drawn from a point to a parabola
Two tangents can be drawn from a point to a parabola
13a. 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)
14. 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)
15. 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
16. 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.
17. Length 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)
18. Pole and Polar
19. some important results at a glance
Ch. 17 Ellipse - Concept Review
1. Introduction
2. 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
But if a
3. Second focus and second directrix of the ellipse
4. Vertices, major and minor axes, foci, directrices and centre of the ellipse
5. Ordinate, double ordinate and latus rectum of the ellipse
6. focal distances of a point on the ellipse
7. 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
8. 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.
9.Parametric equations and parametric coordinates
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
10. 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]
11. 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)
12. Equation of tangent in terms of its slope
13. Equation of tangent at a point
14 Number of tangents drawn from a point to an ellipse
15. Equation of normal in different forms
16 Number of normals
17. Properties of eccentric angles of the conormal points
18. Equation of the pair of tangents from a point to an ellipse
19. Equation of the chord of contacts of tangents
19a. Equation of the chord bisected at a given point
20. Equation of diameter of an ellipse
21. Some properties of ellipse
2. 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
But if a
3. Second focus and second directrix of the ellipse
4. Vertices, major and minor axes, foci, directrices and centre of the ellipse
5. Ordinate, double ordinate and latus rectum of the ellipse
6. focal distances of a point on the ellipse
7. 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
8. 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.
9.Parametric equations and parametric coordinates
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
10. 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]
11. 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)
12. Equation of tangent in terms of its slope
13. Equation of tangent at a point
14 Number of tangents drawn from a point to an ellipse
15. Equation of normal in different forms
16 Number of normals
17. Properties of eccentric angles of the conormal points
18. Equation of the pair of tangents from a point to an ellipse
19. Equation of the chord of contacts of tangents
19a. Equation of the chord bisected at a given point
20. Equation of diameter of an ellipse
21. Some properties of ellipse
Chapter 20 Limits - 1
Material covered in class XIth book.
Limit of a function
Limit x→a f(x) = l, means | f(x)-l| can be made as small as we like by making | x-a| sufficiently small without making x = a.
Since we undertake to make | f(x)-l| as small as required, we will be told how small | f(x)-l| is to be made. We will be given a criterion ε>0 and we must make | f(x)-l| < ε for values of x near a. We must do this by making | x-a| small enough. Hence we must find δ>0 such that when | x-a| < δ (and x≠a), | f(x)-l| will be less than 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.
Note:
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≠0} is called the deleted δ-neighbourhood of a.
Definition of limit
Lim x→a f9x) = 1 if, given any ε>0, we can find δ>0 such that | f(x)-l| < ε whenver 0<| x-a| < δ.
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→ag(x)] provided lim x→a g(x) ≠
0.
(Note: most of the problems given in exercises in the chapter are on applying this condition. Check whether the denominator becomes zero at ‘a’ and then check if the numerator also becomes zero at the ‘a’. If both become zero, it means there is a common factor and one has to remove the common factor to find the limit)
5. If f(x)
Standard limits
1. lim x→a x = a
2. lim x→a kx = ka, k Є R
3. lim x→a xk = ak , k Є R
4. lim x→a k = k
5. lim x→a (xk –ak )/(x-a) = nan-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
Limit of a function
Limit x→a f(x) = l, means | f(x)-l| can be made as small as we like by making | x-a| sufficiently small without making x = a.
Since we undertake to make | f(x)-l| as small as required, we will be told how small | f(x)-l| is to be made. We will be given a criterion ε>0 and we must make | f(x)-l| < ε for values of x near a. We must do this by making | x-a| small enough. Hence we must find δ>0 such that when | x-a| < δ (and x≠a), | f(x)-l| will be less than 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.
Note:
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≠0} is called the deleted δ-neighbourhood of a.
Definition of limit
Lim x→a f9x) = 1 if, given any ε>0, we can find δ>0 such that | f(x)-l| < ε whenver 0<| x-a| < δ.
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→ag(x)] provided lim x→a g(x) ≠
0.
(Note: most of the problems given in exercises in the chapter are on applying this condition. Check whether the denominator becomes zero at ‘a’ and then check if the numerator also becomes zero at the ‘a’. If both become zero, it means there is a common factor and one has to remove the common factor to find the limit)
5. If f(x)
Standard limits
1. lim x→a x = a
2. lim x→a kx = ka, k Є R
3. lim x→a xk = ak , k Є R
4. lim x→a k = k
5. lim x→a (xk –ak )/(x-a) = nan-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
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