# 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(x_{1},y_{1}) and B(x_{2},y_{2}) be
the extremities of a diameter of the circle.

Then the equation of the circle is

(x -x_{1})(x -x_{2}) + (y -y_{1})(y -y_{2}) = 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

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

### Solution

- 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 -

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

- 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. - Determine the equation of a circle whose center is (8, -6) and which passes through the point (5, -2).
- Prove that the points (7, -9) and (11, 3) lie on a circle with center at the origin. Also find its equation.
- 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. - 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.
- 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.
- 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.
- Find the equations of the circles of radius 5 whose centers lie on the x-axis and pass through the point (2,3).
- 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. - Find the equations of circles which touch both the axes and

(i) has radius 3 units

(ii) touch the line x = 2a. - 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.
- 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). - 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.
- 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.
- 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

**1.**(i) x² +y² = 25 (ii) x² +y² -2x +4y = 0

**2.**x² +y² -16x + 12y +75 = 0

**3.**x² +y² -130 = 0

**4.**(i) x² +y² -2ax -2by = 0

(ii) x² +y² -4x +6y -96 = 0

**5.**x² +y² -6x +4y = 0.

**6.**x² +y² +6x -4y -36 = 0

**7.**x² +y² +6y -7 = 0.

**8.**x² +y² -12x +11 = 0, x² +y² +4x -21 = 0

**9.**(i) x² +y² +8y = 0 (ii) x² +y² -6x -8y +16 = 0

**10.**(i) x² +y² ±6x ±6y +9 = 0

(ii) x² +y² -2ax ±2ay +a² = 0

**11.**x² +y² ±6y -16 = 0

**12.**(i) x² +y² -10x ±12y +25 = 0

(ii) x² +y² ±6x +30y = 0

(iii) x² +y² -2x -2y +1 = 0, x² +y² -10x -10y +25 = 0

**13.**x² +y² -10x ±8y +16 = 0

**14.**x² +y² ±10x -6y +9 = 0

**15.**x² +y² ±8x ±10y +16 = 0