The main problem is knowing how to make use of the information that the tangents intersect the coordinate axes at concyclic points. One way would be to use the intersecting chords theorem. This says that if you have a circle, and two lines through a point X, with one line meeting the circle at C and D, and the other line meeting the circle at E and F, then . For this problem, take the two lines to be coordinate axes (so that X will be the origin).

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 .