Circle - Chapter Revision Points

Sections in the chapter

1. Definition
2. Standard equation of a circle
3. Some particular cases of the central form of the equation of a circle
4. General equation of a circle
5. Equation of a circle when the co-ordinates of end points of a diameter are given
6. Intercepts on the axis
7. Position of a point with respect to a circle
8. Equation of a circle in parametric form
9. Intersection of a straight line and a circle
10. The length of the intercept cut off from a line by a circle
11. Tangent to a circle at a given point.
12. Normal to a circle at a given point
13. Length of the tangent from a point to a circle
14. Pair of tangents drawn from a point to a given circle
15. Combined equation of pair of tangents
16. Director circle and its equation
17. Chord of contact of tangents
18. Pole and polar
19. Equation of the chord bisected at a given point
20. Diameter of a circle
21. Common tangents to two circles
22. Common chord of two circles
23. Angle of intersection of two curves and the condition of orthogonality of two circles.
24. Radical axis
25. Equation of a circle through the intersection of a circle and line
26. Circle through the intersection of two circles
27. Coaxial system of circles

Revision Points

Equation of circle in various forms

a. Centre (h,k) and radius a

b. Centre (h,k) and passing through origin

c. Centre (h,k) and circle touches the axis of x

d. Centre (h,k) and circle touches the axis of y

e. Centre (h,k) and and circle touches both axes.

f. general equation

g. circle whose diameter is the line joining two points

h. Circle through three given points

i. Parametric equation of the circle

h. equation of a circle that touches given circle at a given point

j. Equation of a circle passing through the intersection of given circles S1 = 0 and S2 = 0






The standard equation of a circle with center C(h,k) and radius r is as follows:

(x - h)² + (y - k)² = r²





Parametric Equations
The equation of a circle, centred at the origin, is: x2 + y2 = a2, where a is the radius.

Suppose we have a curve which is described by the following two equations:

x = acosθ (1)
y = asinθ (2)

We can eliminate q by squaring and adding the two equations:

x² + y² = a²cos²θ + a²sin²θ = a² .

Hence equations (1) and (2) together also represent a circle centred at the origin with radius a and are known as the parametric equations of the circle. θ is known as the parameter. As θ varies between 0 and 2π, x and y vary.

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Updated  17 Jan 2016, 28 April 2008

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