[物理] 平行光進入拋面後, 所有反射光都會通過焦點的數學證明

前設是反射定律。
Consider a general parabola $y^2=4px$.
The focus of the parabola is $(p, 0)$.

Let a general horizontal line be $y=k$, where $k$ is constant.
It cuts the parabola at $( \frac{k^2}{4p}, k)$.

It can be shown that the general normal line at $( \frac{k^2}{4p}, k)$ is:
$y=-\frac{k}{2p}x+k+\frac{k^3}{8p^2}$

and the reflection of the horizontal line through the normal at $( \frac{k^2}{4p}, k)$ is:
$y = -\frac{4pk}{4p^2 - k^2}x + \frac{k^3}{4p^2 - k^2} + k$, where $k \neq 2p$.

Substitute $(p, 0)$ into $y = -\frac{4pk}{4p^2 - k^2}x + \frac{k^3}{4p^2 - k^2} + k$,

we have
$L.H.S=0$

$R.H.S$ $\\=-\frac{4pk}{4p^2 - k^2}p + \frac{k^3}{4p^2 - k^2} + k \\=\frac{-4p^2k+k^3+4p^2k - k^3}{4p^2 - k^2} \\=0$

$L.H.S=R.H.S$

for $k=2p$, the reflection line is a vertical line pass through $(p,k)$ and $(p,0)$.
Thus, the reflection line passes the focus.

筆者利用Geogebra將反射的情況畫出來了, 可參考
http://tube.geogebra.org/m/2188507