Ellipse and locus of point of intersection of two tangents

Prove that the line lx+my+n=0 is a tangent to the ellipse $\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=0$ if $\displaystyle a^2l^2+b^2m^2=n^2$.

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 $\displaystyle a^2x^2+b^2y^2=a^4+b^4$

I have proved the first part.

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

$\displaystyle y=mx\pm\sqrt{a^2m^2+b^2}$

gradients are related by

$\displaystyle \frac{1}{m_1}.\frac{1}{m_2}=-\frac{b^2}{a^2}$

$\displaystyle m_2=-\frac{a^2}{b^2{m_1}^2}$

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!