Question

Prove that the tangent to the circle $x^2+y^2=5$ at the point $(1,-2)$ also touches the circle $x^2+y^2-8x+6y+20=0$ and find its point of contact.

Answer 1

$T_{(1,-2)}=x-2y-5=0$
Apply $p=r$ with $(4,-3),\sqrt{5}$. It is satisfied. Point of contact as in part (a) is $(3,-1)$.