Problem

The hyperbola $H$ has equation

$$\frac{x^2}{25} - \frac{y^2}{4} = 1.$$

The line $l$ has equation $y = mx + c$, where $m$ and $c$ are constants.

Given that $l$ is a tangent to $H$,

(a) show that $25m^2 = 4 + c^2$.

(b) Hence find the equations of the tangents to $H$ that pass through the point $(1, 2)$.

(c) Find the coordinates of the point of contact each of these tangents makes with $H$.