## Thursday, June 5, 2008

### 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
To satisfy a = k
So equation is
(x-h)²+(y-a)² = a²

(iv)Circle touches the y axis
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.

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.

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)

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

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

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.

(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.