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 (x₁ , y₁) is
   xx₁ +yy₁ +g (x +x₁) +f(y +y₁) +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 (x₁, y₁) is
  (y₁ +f) x -(x₁ +g) y +(g y₁ -f x₁) = 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

1. (i) Find the equation of the tangent to the circle x² + y²= a² at the point P (x₁, y₁) on it.
   (ii) Find the equation of the normal to the circle x² + y² = a² at the point P (x₁, y₁) on it.
2. 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.
3. 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.
4. 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