8. The point \(\mathrm { P } ( 5 \sec \mathrm { u } , 3 \tan \mathrm { u } )\) lies on the hyperbola H with equation \(\frac { \mathrm { x } ^ { 2 } } { 25 } - \frac { \mathrm { y } ^ { 2 } } { 9 } = 1\). The tangent to \(H\) at \(P\) intersects the asymptote of \(H\) with equation \(y = \frac { 3 } { 5 } x\) at the point \(R\) and the asymptote with equation \(\mathrm { y } = - \frac { 3 } { 5 } \mathrm { x }\) at the point S .

1. Use differentiation to show that an equation of the tangent to H at P is $$3 x = 5 y \sin u + 15 \cos u$$
2. Prove that P is the mid-point of RS.