Question 9 - A-Level Maths

$$x ^ { 2 } + 9 y ^ { 2 } = 9$$ The foci of $E$ are $F _ { 1 }$ and $F _ { 2 }$

1. Determine the coordinates of $F _ { 1 }$ and the coordinates of $F _ { 2 }$
2. Write down the equation of each of the directrices of $E$ The point $P$ lies on the ellipse.
Show that $\left| P F _ { 1 } \right| + \left| P F _ { 2 } \right| = 6$ The straight line through $P$ with equation $y = 2 x + c$ meets $E$ again at the point $Q$ The point $M$ is the midpoint of $P Q$
Show that as $P$ varies the locus of $M$ is a straight line passing through the origin.