Prove involving parabolic mirrors

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SUMMARY

The discussion focuses on proving that rays of light traveling parallel to the z-axis converge at a single point when reflected by a parabolic mirror defined by the equation \( z = x^2 + y^2 \). The law of reflection is essential for this proof, which involves analyzing the intersection of light rays with the mirror's surface and their subsequent reflection. The key insight is to utilize vector calculus to derive the equations of the incident and reflected rays, demonstrating their convergence.

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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.
 
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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.

picture-1p.png
 
Last edited:

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