from a point A two tangents are drawn to ellipse (x^2/a^2)+(y^2/b^2)=1 if these tangents intersect the coordinate axis at concyclic points the locus of point P IS (a>b)
Next, let A be the point (p,q). Find the two tangents from A to the ellipse, see where they meet the axes, and use the condition that those four points satisfy the condition of the intersecting chords theorem. That will give you an equation connecting p and q. Finally, replace p and q by x and y to get the equation of the locus of A. I get it to be (part of) the rectangular hyperbola .