Equation of a Circle in different forms

A circle is the locus of a point which moves in a plane so that it remains at a constant distance from a fixed point. The fixed point is called the center and the constant distance is called radius. Radius is always positive.

Standard (or simplest) form

The equation of a circle with O(0,0) as center and r (>0) as radius is
    x² +y² = r²

Central form

The equation of a circle with C(h,k) as center and r (>0) as radius is given by
    (x -h)² +(y -k)² = r²

Diameter form

Let A(x₁,y₁) and B(x₂,y₂) be the extremities of a diameter of the circle.
Then the equation of the circle is
(x -x₁)(x -x₂) + (y -y₁)(y -y₂) = 0
       

Illustrative Examples

Example

Find the equation of a circle whose center is (3,-2) and which passes through the intersection of the lines 5x +7y = 3 and 2x -3y = 7.

Solution

Given lines are
5x +7y -3 = 0    ...(i), and
2x -3y -7 = 0    ...(ii)
Solving (i) and (ii) simultaneously, we get x = 2, y = -1.
So the point of intersection, say P, of the given lines is (2,-1).
Since the center of the circle is C(3,-2) and it passes through the point P(2,-1),
 radius = |CP| = [(2 -3)² +(-1 +2)²] = [1 +1] = 2
Hence the equation of the circle is
    (x -3)² +(y +2)² = (2)²          (central form)
or    x² +y² -6x +4y +11 = 0.

Example

Find the equation of a circle which touches

1. the y-axis at origin and whose radius is 3 units
2. both the co-ordinate axes and the line x = 3.

Solution

1. There are two circles satisfying given conditions. As the circles touch y-axis at the origin, their centers lie on x-axis. Since radius is 3 units, centers of the circles are (3, 0) and (-3, 0) and hence the equations of the circles are
     (x ±3)² +(y -0)² = 3²     or    x² +y² ±6x = 0
2.        
   There are two circles satisfying the given conditions. From above figure, clearly, the centers of these circles are and (3/2,3/2) and (3/2, -3/2) radius of each circle is 3/2. So the required equation is
        x² +y² -3x ±3y + (9/4) = 0
   or  4x² +4y² -12x ±12y +9 = 0

Exercise

1. Find the equation of a circle whose
   (i) center is at the origin and the radius is 5 units.
   (ii) center is (-1, 2) and radius is 5 units.
2. Determine the equation of a circle whose center is (8, -6) and which passes through the point (5, -2).
3. Prove that the points (7, -9) and (11, 3) lie on a circle with center at the origin. Also find its equation.
4. Find the equation of the circle whose
   (i) center is (a, b) and passes through origin.
   (ii) center is (2, -3) and passes through the intersection of the lines 3x -2y -1 = 0 and 4x +y -27 = 0.
5. Find the equation of a circle whose center is the point of intersection of the lines 2x +y = 4 and x -y = 5 and passes through the origin.
6. Find the equation of a circle whose two diameters lie along the lines 2x -3y +12 = 0 and x +4y + 12 = 0 and x +4y -5 = 0 and has area 154 square units.
7. Find the equation of the circle whose center lies on the negative direction of y-axis at a distance 3 units from origin and whose radius is 4 units.
8. Find the equations of the circles of radius 5 whose centers lie on the x-axis and pass through the point (2,3).
9. Find the equation of the circle
   (i) whose center is (0, -4) and which touches the x-axis.
   (ii) whose center is (3, 4) and touches the y-axis.
10. Find the equations of circles which touch both the axes and
    (i) has radius 3 units
    (ii) touch the line x = 2a.
11. Find the equations of circles which pass through two points on x-axis at distances of 4 units from the origin and whose radius is 5 units.
12. Find the equations of circles
    (i) which touch the x-axis on the positive direction at a distance 5 units from the origin and has radius 6 units.
    (ii) passing through the origin, radius 17 and ordinate of the center is -15.
    (iii) which touch both the axes and pass through the point (2, 1).
13. Find the equations of circles which touch the y-axis at a distance of 4 units from the origin and cut off an intercept of 6 units along the positive direction of x-axis.
14. Find the equations to the circles touching axis of y at the point (0, 3) and making an intercept of 8 units on x-axis.
15. Find the equations to the circles which touch the x-axis at a distance of 4 units from the origin and cut off an intercept of 6 from the y-axis.

Answers