How to find another solution for two cone intersection? Lets say we have two cones and they intersect:
$$
\sqrt{(x-a_1)^2 + (y-a_2)^2} + a_3 = \sqrt{(x-b_1)^2+(y-b_2)^2} + b_3
$$
And we know one solution $x_1, y_1$ for this intersection.
Is there any simple way to find another solution $x_2, y_2$ for the equation, using only the first solution?
Edit: here is example how the intersection of two cones looks like in this case:


The intersection is a hyperbola, symmetric with respect to the vertical plane passing through the cones vertices. It follows that given $(x_1,y_1)$ you can find another point $(x_2,y_2)$ on the hyperbola by taking the symmetric of $(x_1,y_1)$ with respect to the line passing through $(a_1,a_2)$ and $(b_1,b_2)$:
$$
(x_2,y_2)=2{(x_1-a_1)(b_1-a_1)+(y_1-a_2)(b_2-a_2)
\over\sqrt{(b_1-a_1)^2+(b_2-a_2)^2}}(b_1-a_1,b_2-a_2)+(2a_1-x_1,2a_2-y_1).
$$
