ODE modeling a curved reflective surface, describe shape of curve

1. Sep 16, 2009

Breedlove

1. The problem statement, all variables and given/known data
Assume that when the curve C shown below is revolved about the x-axis, it generates a surface of revolution with the property that all light rays L parallel to the x-axis striking the surface are reflected to a single point O (the origin). Use the fact that the angle of incidence is equal to the angle of reflection to determine a differential equation that describes the shape of the curve C.

(I drew up the diagram in MSpaint, but can't figure out how to upload it. HELP!)

2. Relevant equations

It's a model problem, so I assume that there is going to be some sort of proportionality constant, but as for equations I'm not sure at this point.

3. The attempt at a solution
Because I can't figure out how to upload the figure, I will attempt to describe what's happening and how far I was able to go. There is a curve C and a tangent line to C at the point where light hits C and reflects. The angle of incident, the angle at which the light hits C, is also the angle at which the light leaves C (theta), as measured from the tangent line, NOT C. The obtuse angle that the light makes with the x-axis is phi, which equals 2(theta) because of alternate interior angles.
Okay. There is little likelihood that I would be able to follow all this without a diagram, but I hope it's not too lost. I think I should just get this question out there so someone can enlighten me so I can better describe the problem. There was a hjint that once you figured out that phi=2(theta) you could use "an appropriate tironometric identity". What kind of identitiy could give me an idea about the shape of C? I'm pretty sure it's parabolic, although my picture doesn't really look like that, but it's paint. Okay. Thanks!

okay I attached it, I hope it works!

Okay, in the attached picture, the thetas are supposed to be the angle between the tangent line and the path of the light ray, not the angle between the curve C and the light ray, sorry about that.

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Last edited: Sep 16, 2009
2. Sep 16, 2009

LCKurtz

I haven't worked it all out but here's where I might start. Label the point where the light hits on the curve (x,y). You know the slope of the curve:

$$y' = \tan(\theta)$$

and from the picture

$$\frac y x = \tan(\phi)$$

and you know

$$\phi = 2\theta$$

Take that and run with it.