9 An ellipse is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{cf9337b9-b766-4ce5-967c-5d7522e2aa42-5_453_633_365_699}
The ellipse intersects the \(x\)-axis at the points \(A\) and \(B\). The equation of the ellipse is
$$\frac { ( x - 4 ) ^ { 2 } } { 4 } + y ^ { 2 } = 1$$
- Find the \(x\)-coordinates of \(A\) and \(B\).
- The line \(y = m x ( m > 0 )\) is a tangent to the ellipse, with point of contact \(P\).
- Show that the \(x\)-coordinate of \(P\) satisfies the equation
$$\left( 1 + 4 m ^ { 2 } \right) x ^ { 2 } - 8 x + 12 = 0$$
- Hence find the exact value of \(m\).
- Find the coordinates of \(P\).