Find locus of a circle which meets four lines intersections

Question

Find equation of circle which meets the lines $y=\pm mx, y=\pm c$ and circle $x^2+y^2=a^2$.

After writing the joint equations of the given lines unable to relate it with equation of circle involving radius $a$.

Answer 1

Given for a 2d situation. If you want for 3d you need to edit your question accordingly.

I have taken $m=2, c=2$ for the example shown. The circle has equation $$ x^2+y^2= \big(\sqrt{c^2+c^2/m^2}\big)^2$$