Prove involving parabolic mirrors

[alejandro7]

I'm having trouble with the following problem:

"Consider the parabolic mirror given by the equation $z=x^2+y^2$. Show that when the rays of light that travel parallel to the $z$ axis pass through the same point when reflected."

I'm familiar with the law of reflection but I'm stuck because I don't know how to apply vector calculus to this situation. I have tried to apply the law of reflection but I can't seem to get anywhere.

Thanks.

 

[maajdl]

You simply need to consider the equations of two straight lines:

- a certain ray parallel to the axis, and see where it cuts the mirror
- the ray reflected from there, according the the law of reflexion

The picture below suggests what you have to do: simply follow the red path.