8. The ellipse \(E\) has equation
$$\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$$
- Determine the eccentricity of \(E\)
- Hence, for this ellipse, determine
- the coordinates of the foci,
- 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 \(P\)
- Using calculus, show that an equation for \(l _ { 1 }\) is
$$2 x \cos \theta + 3 y \sin \theta = 6$$
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 \(Q\) - Determine the coordinates of \(Q\)
- Show that, as \(\theta\) varies, the point \(Q\) lies on the curve with equation
$$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = \alpha x ^ { 2 } + \beta y ^ { 2 }$$
where \(\alpha\) and \(\beta\) are constants to be determined.
\includegraphics[max width=\textwidth, alt={}]{cfc4afbd-3353-4f9f-b954-cb5178ebcf6c-36_2817_1962_105_105}