We have seen the word tangent before when we studied parabolas. We defined it loosely as a line that just touches a curve.

A normal line is perpendicular to the tangent.

Here is how a tangent and normal to a circle would look. Move the centre $C$ and the point $P$ to get a sense for how the tangent and normal behave.

In this tutorial, we are going to see how to work out the equation of a tangent and the equation of a normal using the point-slope formula:

1. The point $P(7,-8)$ lies on the circle with equation

   $$(x-5{)}^2+(y+10{)}^2=8$$

   Find the equation of the normal to the circle at $P.$

2. The point $Q(-5,-1)$ lies on the circle

   $$(x-5{)}^2-1=(10-y)(10+y)$$

   Find an equation for the tangent to the circle at the point $Q.$