Do all light rays reflecting off slope field mirrors pass through (0,k)?

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bomba923
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Ok...well, here goes:
Let's say we draw a slope-field for

[tex]\frac{{dy}}{{dx}} = \frac{x}{{k - y + \sqrt {x^2 + \left( {k - y} \right)^2 } }}[/tex]

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Now, let's say a ray of light traveling in the direction [tex]\left\langle {0, - 1} \right\rangle[/tex] "hits" one of the slope segments we drew. However, each "slope segment" is a actually a short planar mirror!

And so,

*Regardless of which segment this light ray strikes, will the light ray be reflected through the point [tex](0,k)[/tex] ?
(assuming this ray strikes one and only one "mirror"/slope-segment)
 
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Rewrite the equation for dy/dx taking dy/dx=C where C is an arbitrary constant, what curve the resulting equation represents? What properties has such a "mirror"? To what point the ray will be focused?
 
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