Problem

The ellipse $E$ has equation

$$x^2+9y^2=9$$

The foci of $E$ are $F_1$ and $F_2$

(a) (i) Determine the coordinates of $F_1$ and the coordinates of $F_2$

(ii) Write down the equation of each of the directrices of $E$

(b) Show that $PF_1\times PF_2=6$

The straight line through $P$ with equation $y=2x+c$ meets $E$ again at the point $Q$

The point $M$ is the midpoint of $PQ$

(c) Show that as $P$ varies the locus of $M$ is a straight line passing through the origin.