Problem

The ellipse $E$ has equation

$$\frac{x^2}{9}+\frac{y^2}{4}=1$$

(a) Determine the eccentricity of $E$

(b) Hence, for this ellipse, determine

(i) the coordinates of the foci,

(ii) the equations of the directrices.

The point $P$ lies on $E$ and has coordinates $(3\cos\theta,2\sin\theta)$.

The line $l_1$ is the tangent to $E$ at the point $P$

(c) Using calculus, show that an equation for $l_1$ is

$$2x\cos\theta+3y\sin\theta=6$$

The line $l_2$ passes through the origin and is perpendicular to $l_1$

The line $l_1$ intersects the line $l_2$ at the point $Q$

(d) Determine the coordinates of Q

(e) Show that, as $\theta$ varies, the point $Q$ lies on the curve with equation

$$(x^2+y^2)^2=\alpha x^2+\beta y^2$$

where $\alpha$ and $\beta$ are constants to be determined.