Tangent lines to circle $ x^2+y^2=100 $ passing point (14, 2)

Question

I'm asked to give tangent lines to $ x^2+y^2=100 $, so that both tangent lines go through the point $ (14,2)$.

Implicit differentiation gives:

$dy/dx=-x/y$

While graphing I noted that the circle , doesn't go through (14,2).

I constructed the following tangent line to the point (14,2):

$ y-2=-7(x-14) $

but it's not tangent to the circle.

I'm a bit stuck, need some help 😀

Answer 1

We form the system $$\begin{cases}y-2=m(x-14),\\x^2+y^2=100\end{cases}$$

or, by elimination

$$x^2+(m(x-14)+2)^2=100.$$

This equation has a double root (hence the line is tangent) when the delta cancels, i.e. when

$$12m^2-7m-12=0.$$