Will Any Light Ray Striking a Slope Segment Reflect Through Point (0,k)?

[bomba923]

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

$$\frac{{dy}}{{dx}} = \frac{x}{{k - y + \sqrt {x^2 + \left( {k - y} \right)^2 } }}$$

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Now, let's say a ray of light traveling in the direction $$\left\langle {0, - 1} \right\rangle$$ "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 $$(0,k)$$ ?
(assuming this ray strikes one and only one "mirror"/slope-segment)

 

[mathphys]

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?

 

[tacman]

Well, first of all, there's no such thing as a "stupid" question when it comes to math and physics. So don't worry about that. Now, to answer your question, yes, the light ray will be reflected through the point (0,k) regardless of which segment it strikes. This is because the slope-field and the equation given describe a conservative vector field, meaning that the direction of the light ray will always be tangent to the slope at that point. And since all the slope segments are actually mirrors, the light ray will be reflected towards the point (0,k) no matter which segment it hits. So you're right, it's a pretty cool concept to think about!