Ellipse and locus of point of intersection of two tangents

Prove that the line lx+my+n=0 is a tangent to the ellipse if .

Tangents are drawn from a point P to the ellipse. Lines from the origin are drawn perpendicular to the tangents. If these lines are conjugate diameters, prove that the equation of the locus of P is

I have proved the first part.

For this next part the equation of a tangent to an ellipse in the form

gradients are related by

I used the two gradients, m, in two equations for tangents and tried to solve for x and y, am I doing it right? because I get very very long and difficult expressions.

Thanks!