Question 8 - A-Level Maths

8 A curve has equation $y ^ { 2 } = 12 x$.

Sketch the curve.
1. The curve is translated by 2 units in the positive $y$ direction. Write down the equation of the curve after this translation.
2. The original curve is reflected in the line $y = x$. Write down the equation of the curve after this reflection.
1. 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 $$x ^ { 2 } + ( 2 c - 12 ) x + c ^ { 2 } = 0$$
2. Hence find the value of $c$ for which the straight line is a tangent to the curve.
3. Using this value of $c$, find the coordinates of the point where the line touches the curve.
4. In the case where $c = 4$, determine whether the line intersects the curve or not.