9 The diagram shows the parabola \(y ^ { 2 } = 4 x\) and the point \(A\) with coordinates \(( 3,4 )\).
\includegraphics[max width=\textwidth, alt={}, center]{504b79bf-1bcc-4fa7-a7a0-689c21a8b03a-05_732_657_370_689}
- Find an equation of the straight line having gradient \(m\) and passing through the point \(A ( 3,4 )\).
- Show that, if this straight line intersects the parabola, then the \(y\)-coordinates of the points of intersection satisfy the equation
$$m y ^ { 2 } - 4 y + ( 16 - 12 m ) = 0$$
- By considering the discriminant of the equation in part (b), find the equations of the two tangents to the parabola which pass through \(A\).
(No credit will be given for solutions based on differentiation.) - Find the coordinates of the points at which these tangents touch the parabola.