A light ray comes down the line

A light ray comes down the line $x=a$. It hits the parabolic reflector $y=x^2$ at $P=(a,a^2)$.
(a) Find the tangent line at $P$. Locate the point $Q$ where that line crosses the $y$ axis.
(b) Check that $P$ and $Q$ are the same distance from the focus at $F=(0,\frac{1}{4})$.
(c) Show from (b) that the figure has equal angles.
(d) What law of physics makes every ray reflect off the parabola to the focus at $F$?

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §2.3 The Slope and the Tangent Line, #25