$\begin{cases} \dfrac{x}{\sqrt{x^2+(y-y_A)^2}} + \dfrac{x-x_B}{\sqrt{(x-x_B)^2+(y-y_B)^2}} = 2x\lambda \\\\ \dfrac{y - y_A}{\sqrt{x^2+(y-y_A)^2}} + \dfrac{y-y_B}{\sqrt{(x-x_B)^2+(y-y_B)^2}} = 2y\lambda \\\\ x^2+y^2=R^2 \end{cases}$
How can I solve the above system for $x,y,\lambda$?
$y_A,x_B,y_B$ and $R$ are given constants.
I tried squaring and adding the two first lines, but got nothing. Substituting the third line into the two first lines also didn't solve it for me.
If the line segment $AB$ intersects the circle, let $P$ be any of the points where it does. If it doesn't intersect the circle, you want the point $P$ on the circle where a ray of light from $A$ will reflect to $B$. This has the property that the ray from the centre of the circle through $P$ bisects the angle $APB$.