Let a conic section in 2D : $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$ For a circle with radius $R, B=D=E=0$ and $F=R2, A=B=1$. The conic section may be assumed a mirror.
Let us assume some optical rays originating from the coordinates $(0,0)$. For a circle, all rays come back to the point $(0,0)$. For a parabola, this is more spread around this point. So let us assume that a ray is hitting the point $(x_0,y_0)$ of the conic section and comes back. What is the equation of this reflected ray as a function of $(A,B,C,D,E,F)$ and $(x_0,y_0)$ ? And, how to derive an expression for all the rays coming back at $x=0$ and $y∈[a,b]$, for $x_0>0,y_0∈[−r,r]$ varying over the mirror ?