# How to find equation of two tangents at 2 points on a circle?

There is a circle given with equation (x-2)^2 + (y-3)^2 = 20.

There are two points that lie on the circumference of the circle which I have worked out, P(-2,5) and Q(4,7).

The diagram of the circle does not contain any tangents drawn on, and there is only a line ( line L) which intersects the circle at P and Q. The question states : " Find the equation of the tangent at the points P and Q."

The diagram of the circle does not contain any tangents drawn on, and there is only a line ( line L) which intersects the circle at P and Q.

I am stuck on this question and any and all help would be greatly appreciated.

• Are you supposed to make use of $L$ in some way, or is it just there for decoration? – amd Nov 2 '17 at 23:55

If $C$ is the center of the circle, and $P$ is a point on the circle, then the tangent line to the circle at $P$ has slope equal to the negative reciprocal of the line through $C$ and $P$. So, find the slope of the line through $C$ and $P$, take the negative reciprocal of that, and that is the slope of the line you seek. Since you also know it passes through $P$, you have enough information to find the equation of the line.
Do the same with $Q$.