Through any point P of the hyperbola $\displaystyle \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, a line QPR is drawn with a fixed gradient m, meeting the asymptotes in Q & R. Show that the product $\displaystyle \text{QP}\cdot\text{PR} = \frac{a^2b^2(1 + m^2)}{b^2 - a^2m^2}$.