Solve 2nd Order ODE Mirror for Parallel Reflection from Origin

[Design]

Homework Statement

A curved mirror of equation y=y(x) has that property that whenever a ray of light emanates from the origin it reflects parallel to the x-axis. Find the equation of the mirror


Don't even know how to get started on this, Don't need a solution just need some starting hints / help!

thank you

 

[HallsofIvy]

Let y= f(x) be the equation of the mirror. At point $(x_0, f(x_0))$, the tangent line to the curve is given by [itex]y= f'(x_0)(x- x_0)+ f(x_0)[/tex]. The line parallel to the x-axis has equation $y= f(x_0)$ and the equation of the line from the origin to the point on the curve has $y= f(x_0)x/x_0$.<br /> <br /> To "reflect from the mirror" those two lines must make equal angles with the tangent line.[/itex]

 

[Design]

Let y= f(x) be the equation of the mirror. At point $(x_0, f(x_0))$, the tangent line to the curve is given by [itex]y= f'(x_0)(x- x_0)+ f(x_0)[/tex]. The line parallel to the x-axis has equation $y= f(x_0)$ and the equation of the line from the origin to the point on the curve has $y= f(x_0)x/x_0$.<br /> <br /> To "reflect from the mirror" those two lines must make equal angles with the tangent line.[/itex]

[itex] <br /> How do I come up with this information my self and how do you know the the origin to the point on the curve has equation $y= f(x_0)x/x_0$?[/itex]

 

[Design]

Bump!, I understand how you got the equation at the point to the curve now.
I don't understand how to use these three equations to make an ODE. I know that the angles must be equal but I don't understand how they relate to the question. I also know I can take a tangent line at the origin, to the intersection of y = $y= f(x_0)$ and that would be a equation that has the same slope has my tangent line at $x_0$

 

[Design]

bump! same question still