# Tangent and Normal to a Circle at a Point

### The equation of the tangent at a point on a circle

The equation of the tangent to the circle x² +y² +2 g x +2 f y +c = 0 at
the point P (x_{1} , y_{1}) is

xx_{1} +yy_{1} +g (x +x_{1}) +f(y +y_{1}) +c = 0

### The equation of the normal at a point on the circle

The equation of the normal to the circle x² +y² +2 g x +2 f y +c = 0 at the
point P (x_{1}, y_{1}) is

(y_{1} +f) x -(x_{1} +g) y +(g y_{1} -f x_{1}) = 0

Normal at a point on the circle passes through the center of the circle.

## Illustrative Examples

### Example

Find the equations of tangent and normal to the circle x² +y² -5 x +2 y +3 = 0 at the point (2, -3).

### Solution

The given circle is x² +y² -5 x +2 +2 y +3 = 0 ... (i)

The equation of the tangent to the circle (i) at the point P (2, - 3) is

x. 2 + y. (-3) -5.(1/2).(x +2) +2.(1/2).(y/3) +3 = 0

=> 4 x -6 y -5 x -10 +2 y -6 +6 = 0

=> -x -4 y -10 = 0 => x +4 y +10 = 0

The slope of the tangent at P = - 1/4

=> the slope of the normal at P = 4

The equation of the normal to the circle (i) at P (2, -3) is

y +3 = 4 (x - 2) i.e. 4x - y -11 = 0

## Exercise

- (i) Find the equation of the tangent to the circle x² +
y²= a² at the point P (x
_{1}, y_{1}) on it.

(ii) Find the equation of the normal to the circle x² + y² = a² at the point P (x_{1}, y_{1}) on it. - Find the equations of the tangent and the normal to the
following circles at the given points.

(i) x² +y² = 169 at (12, - 5)

(ii) 4 x² +4 y² = 25 at (3/2, -2)

(iii) x² + y² -4 x +2 y +3 = 0 at (1, -2)

(iv) 3 x² +3 y² -4 x -9 y = 0 at the origin. - Find the equations of the tangent and the normal to the
following circles:

(i) x² +y² = 10 at the points whose abscissa is 1.

(ii) x² +y² -8 x -2 y +12 = 0 at the points whose ordinate is -1. - Show that the tangents drawn at the points (12, - 5) and (5, 12) to the circle x² + y² = 169 are perpendicular to each other.

## Answers

**1.**(i) x x

_{1}+y y

_{1}= a² (ii) y

_{1}x -x

_{1}y = 0

**2.**(i)12 x -5 y -169 = 0; 5x +12 y = 0

(ii) 6x +8 y +25 = 0; 4x -3y = 0

(iii) x +y +1 = 0; x -y -3 = 0

(iv) 4 x + 9 y = 0; 9 x -4 y = 0

**3.**(i) x +3 y -10 = 0, x -3 y -10 = 0, 3 x -y = 0,

(ii) x -2 y -7 = 0, x +2 y -1 = 0, 2 x +y -9 = 0, 2 x - y -7 = 0