Derive ordinary differential equation in physical modeling of a searchlight reflector

Question

What shape should the reflector (a rotating surface) of the searchlight have to be, so that the light beam from the point source can be reflected as parallel wire bundles？

This problem is just about an ODE, and my biggest problem is that I don't know how to construct differential equations from physics. The answer provided by the author is

$$y^2=2C\left( x+\frac{1}{2} C\right)$$

Answer 1

According to the diagram, to achieve parallel reflection, the slope at (x,y) has to be perpendicular to that of the angle bisector line whose slope is $-\tan(\alpha/2)$. So, the following equations can be established,

$$\frac{dy}{dx} = \frac{1}{\tan\alpha /2}, \space \space \space \tan\alpha = -\frac{y}{x}$$

Next, use the identity

$$\tan\alpha = \frac{2\tan\alpha/2}{1-\tan^2\alpha/2}$$

to arrive at the following ODE

$$-\frac{y}{x} = \frac{2\frac{dx}{dy}}{1-\left(\frac{dx}{dy}\right)^2}$$

which has the solution $y^2 = 2C(x+C/2)$.

Answer 2

$$\because \ number \ of\ constants\ =1\ \ \ \\ \\ \\ \because y^2=2cx+c^2\ \ (*)\ \ \ \ , by\ diff\\ \\ \\ \therefore 2yy'=2c\Rightarrow yy'=c\\ \\ \\ by \ substitution\ in\ (*)\ \ \ then\ we\ have\ \ \ \\ \\ \\ \therefore \ y^2=2x(yy')+y^2\ y'^2\\ \\ \\ \\ \therefore \ y=2xy'+yy'^2$$