8 A curve has equation $y ^ { 2 } = 12 x$.\\
(a) Sketch the curve.\\
(b) (i) The curve is translated by 2 units in the positive $y$ direction. Write down the equation of the curve after this translation.\\
(ii) The original curve is reflected in the line $y = x$. Write down the equation of the curve after this reflection.\\
(c) (i) Show that if the straight line $y = x + c$, where $c$ is a constant, intersects the curve $y ^ { 2 } = 12 x$, then the $x$-coordinates of the points of intersection satisfy the equation

$$x ^ { 2 } + ( 2 c - 12 ) x + c ^ { 2 } = 0$$

(ii) Hence find the value of $c$ for which the straight line is a tangent to the curve.\\
(iii) Using this value of $c$, find the coordinates of the point where the line touches the curve.\\
(iv) In the case where $c = 4$, determine whether the line intersects the curve or not.

\hfill \mbox{\textit{AQA FP1  Q8}}