Chapter 2.P, Problem 18P

Let $P\left(x_1,y_1\right)$ be a point on the parabola $y^2=4px$ with focus $F\left(p,0\right).$ Let $\alpha$ e the angle between the parabola and the line segment FP, and let $\beta$ be the angle between the horizontal line $y=y_1$ and the parabola as in the figure. Prove that $\alpha =\beta$. (Thus, by a principle of geometrical optics, light from a source placed at F will be reflected along a line parallel to the x-axis. This explains why paraboloids, the surfaces obtained by rotating parabolas about their axes, are used as the shape of some automobile headlights and mirrors for telescopes.)