The point of intersection of the tangents at the point P on the ellipse x^2/a^2+y^2/b^2=1(standard ellipse) and its corresponding point Q on the auxiliary circle lies on the line:

Let P (acosΘ , bsinΘ)
so Q (acosΘ , asinΘ)

Now write Eqn of tangent and then find intersection point
Eqn of tangent to ellipse at P : xcosΘ / a + ysinΘ /b = 1
Eqn of tangent to circle at Q : xcosΘ + ysinΘ = a

Now solve and find intersection point.

Thanks