- The ellipse \(E\) has equation
$$\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$$
The line \(l\) has equation \(y = k x - 3\), where \(k\) is a constant.
Given that \(E\) and \(l\) meet at 2 distinct points \(P\) and \(Q\)
- show that the \(x\) coordinates of \(P\) and \(Q\) are solutions of the equation
$$\left( 9 k ^ { 2 } + 4 \right) x ^ { 2 } - 54 k x + 45 = 0$$
The point \(M\) is the midpoint of \(P Q\)
- Determine, in simplest form in terms of \(k\), the coordinates of \(M\)
- Hence show that, as \(k\) varies, \(M\) lies on the curve with equation
$$x ^ { 2 } + p y ^ { 2 } = q y$$
where \(p\) and \(q\) are constants to be determined.