Question 9 - A-Level Maths

9 A parabola $P$ has equation $y ^ { 2 } = x - 2$.

1. Sketch the parabola $P$.
2. On your sketch, draw the two tangents to $P$ which pass through the point $( - 2,0 )$.
1. Show that, if the line $y = m ( x + 2 )$ intersects $P$, then the $x$-coordinates of the points of intersection must satisfy the equation $$m ^ { 2 } x ^ { 2 } + \left( 4 m ^ { 2 } - 1 \right) x + \left( 4 m ^ { 2 } + 2 \right) = 0$$
2. Show that, if this equation has equal roots, then $$16 m ^ { 2 } = 1$$
3. Hence find the coordinates of the points at which the tangents to $P$ from the point $( - 2,0 )$ touch the parabola $P$.