(a) Check that the line … is tangent to the circle …

This problem is about parabolas, are about ellipses, are about hyperbolas.

(a) Check that the line $x{x}_0+y{y}_0=r^2$ is tangent to the circle $x^2+y^2=r^2$ at $(x_0,y_0)$.
(b) For the ellipse $x^2/a^2+y^2/b^2=1$ show that the tangent equation is $x{x}_0/a^2+y{y}_0/b^2=1$. (Check the slope.)

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §3.5 Parabolas, Ellipses, and Hyperbolas, #28