5. The ellipse \(E\) has equation
$$x ^ { 2 } + 9 y ^ { 2 } = 9$$
The point \(P ( a \cos \theta , b \sin \theta )\) is a general point on the ellipse \(E\).
- Write down the value of \(a\) and the value of \(b\).
The line \(L\) is a tangent to \(E\) at the point \(P\).
- Show that an equation of the line \(L\) is given by
$$3 y \sin \theta + x \cos \theta = 3$$
The line \(L\) meets the \(x\)-axis at the point \(Q\) and meets the \(y\)-axis at the point \(R\).
- Show that the area of the triangle \(O Q R\), where \(O\) is the origin, is given by
$$k \operatorname { cosec } 2 \theta$$
where \(k\) is a constant to be found.
The point \(M\) is the midpoint of \(Q R\).
- Find a cartesian equation of the locus of \(M\), giving your answer in the form \(y ^ { 2 } = \mathrm { f } ( x )\).