9 The diagram shows the hyperbola $$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$ and its asymptotes.
\includegraphics[max width=\textwidth, alt={}, center]{3c141dcb-4a5e-45ff-9c8e-e06762c03d10-6_798_939_612_555} The constants \(a\) and \(b\) are positive integers.
The point \(A\) on the hyperbola has coordinates ( 2,0 ).
The equations of the asymptotes are \(y = 2 x\) and \(y = - 2 x\).

1. Show that \(a = 2\) and \(b = 4\).
2. The point \(P\) has coordinates ( 1,0 ). A straight line passes through \(P\) and has gradient \(m\). Show that, if this line intersects the hyperbola, the \(x\)-coordinates of the points of intersection satisfy the equation $$\left( m ^ { 2 } - 4 \right) x ^ { 2 } - 2 m ^ { 2 } x + \left( m ^ { 2 } + 16 \right) = 0$$
3. Show that this equation has equal roots if \(3 m ^ { 2 } = 16\).
4. There are two tangents to the hyperbola which pass through \(P\). Find the coordinates of the points at which these tangents touch the hyperbola.
   (No credit will be given for solutions based on differentiation.)