Solution

We show that a parabolic surface will collect all light rays parallel to its axis of symmetry to its focal point as follows. Suppose that the equation of the parabola in the figure is $y^2=4cx$, so its focal point is at $F(c,0)$. Let $P(a^2/(4c),a)$ be any point on the parabola. (In the figure, $a$ is negative.) Then the slope of the tangent line (shown in blue) at $P$ is

$$\left.\frac{dy}{dx}\right|_{y=a}=\frac{2c}{a},$$

and the slope of the normal line (shown in green) at $P$ is $-a/(2c)$. The slope of $PF$ is

$$\tan\angle APF=\frac{a}{a^2/(4c)-c}=\frac{2(-a/(2c))}{1-(-a/(2c))^2)}=\tan(2\angle APB.)$$

In other words, the red lines obey the law of reflection, so an arbitrary light ray parallel to the axis of symmetry of the parabola is collected at its focal point.