Finding Point Q in a Parabolic Dish: Proving the Reflection Theory

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SUMMARY

The discussion centers on proving that all reflecting light rays hitting a parabolic dish defined by the equation z=c(x^2+y^2) converge at a single point Q along the Z-axis. The user initially attempted to simplify the problem by reducing it to two dimensions with the parabola y=ax^2 but struggled to determine the correct angle of incidence for the rays. The need for a comprehensive proof and the identification of point Q are emphasized, with a suggestion that the incident rays must be parallel to the Z-axis for the proof to hold true.

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gipc
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I have the following "Parabolic Dish" z=c(x^2+y^2)
I have to prove that all the reflecting light rays that hit that dish go through the same point Q in the Z axis, and then I have to find said point Q.


I've thought of reducing the problem to 2 dimensions. Started with the parabola y = ax^2. Tried to find the angle that some line x = c strikes the curve but couldn't find the correct angle. Maybe do some gradient magic?


I'm unsure really, how to set it all together for a good proof, and how to finally find the point Q.

Please help :)
 
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gipc said:
I have the following "Parabolic Dish" z=c(x^2+y^2)
I have to prove that all the reflecting light rays that hit that dish go through the same point Q in the Z axis, and then I have to find said point Q.


I've thought of reducing the problem to 2 dimensions. Started with the parabola y = ax^2. Tried to find the angle that some line x = c strikes the curve but couldn't find the correct angle. Maybe do some gradient magic?

I'm unsure really, how to set it all together for a good proof, and how to finally find the point Q.

Please help :)
I doubt that this is the complete problem as given. The incident rays would all need to be parallel to the z-axis.
 


Ohh, yes, obviously they are indeed parallel :)
 

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