Problem

4. A rectangular hyperbola $H$ has equation $xy = 25$

The point $P\left(5t, \dfrac{5}{t}\right)$, $t \neq 0$, is a general point on $H$.

(a) Show that the equation of the tangent to $H$ at $P$ is $t^2 y + x = 10t$

The distinct points $Q$ and $R$ lie on $H$. The tangent to $H$ at the point $Q$ and the tangent to $H$ at the point $R$ meet at the point $(15, -5)$.

(b) Find the coordinates of the points $Q$ and $R$.