Problem

The ellipse $E$ has equation

$$\frac{x^2}{49}+\frac{y^2}{b^2}=1$$

where $b$ is a constant and $0<b<7$

The eccentricity of the ellipse is $e$

(a) Write down, in terms of $e$ only,

(i) the coordinates of the foci of $E$

(ii) the equations of the directrices of $E$

Given that

• the point $P(x,y)$ lies on $E$ where $x>0$

• the point $S$ is the focus of $E$ on the positive $x$-axis

• the line $l$ is the directrix of $E$ which crosses the positive $x$-axis

• the point $M$ lies on $l$ such that the line through $P$ and $M$ is parallel to the $x$-axis

(b) determine an expression for

(i) $PS^2$ in terms of $e$, $x$ and $y$

(ii) $PM^2$ in terms of $e$ and $x$

(c) Hence show that

$$b^2=49(1-e^2)$$

Given that $E$ crosses the $y$-axis at the points with coordinates $\left(0,\pm 4\sqrt3\right)$

(d) determine the value of $e$

Given that the $x$ coordinate of $P$ is

$$\frac72$$

(e) determine the area of triangle $OPM$, where $O$ is the origin.