How to find the value of L in the figure?

Question

How do I find the value of L in the problem below?

Answer 1

Let's call the angle between the red lines $2\beta$. Then the line connecting the reflection point to the origin is the bisector of the reflection angle. In the triangle origin, center of circle, reflection point, you write the law of sines: $$\frac{\sin\alpha}R=\frac{\sin\beta}{c_x}$$ Then in the triangle, reflection point, center of circle, $(L,0)$ you have the angle at center $\alpha+\beta$, so the angle at $L$ is $180^\circ-(\alpha+\beta)-\beta$. Once again, write law of sines: $$\frac{\sin(180^\circ-\alpha-2\beta)}R=\frac{\sin\beta}{L-c_x}$$