- The point \(P\) lies on the ellipse \(E\) with equation
$$\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 9 } = 1$$
\(N\) is the foot of the perpendicular from point \(P\) to the line \(x = 8\)
\(M\) is the midpoint of \(P N\).
- Sketch the graph of the ellipse \(E\), showing also the line \(x = 8\) and a possible position for the line \(P N\).
- Find an equation of the locus of \(M\) as \(P\) moves around the ellipse.
- Show that this locus is a circle and state its centre and radius.