Question:

The tangents to the ellipse with equation $\displaystyle \frac{x^{2}}{a^2} + \frac{y^{2}}{b^2} = 1$ at the points $\displaystyle P(acost,bsint)$ and $\displaystyle Q(-asint,bcost)$ intersect at the point $\displaystyle R$. As $\displaystyle t$ varies, show that $\displaystyle R$ lies on the curve with equation $\displaystyle \frac{x^{2}}{a^2} + \frac{y^{2}}{b^2} = 2$.