Chord midpoint locus

1. 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 \(Q\)

1. 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 \(P Q\)
2. Determine, in simplest form in terms of \(k\), the coordinates of \(M\)
3. 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.

1. Given that \(y = \operatorname { artanh } \frac { x } { \sqrt { } \left( 1 + x ^ { 2 } \right) }\)
   show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { } \left( 1 + x ^ { 2 } \right) }\)
   
2. \hspace{0pt} [In this question you may use the appropriate trigonometric identities on page 6 of the pink Mathematical Formulae and Statistical Tables.]

The points \(P ( 3 \cos \alpha , 2 \sin \alpha )\) and \(Q ( 3 \cos \beta , 2 \sin \beta )\), where \(\alpha \neq \beta\), lie on the ellipse with equation $$\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$$

1. Show the equation of the chord \(P Q\) is $$\frac { x } { 3 } \cos \frac { ( \alpha + \beta ) } { 2 } + \frac { y } { 2 } \sin \frac { ( \alpha + \beta ) } { 2 } = \cos \frac { ( \alpha - \beta ) } { 2 }$$
2. Write down the coordinates of the mid-point of \(P Q\). Given that the gradient, \(m\), of the chord \(P Q\) is a constant,
3. show that the centre of the chord lies on a line $$y = - k x$$ expressing \(k\) in terms of \(m\).

1. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\) where \(c\) is a positive constant.

The line \(l\) has equation \(x - 2 y = c\)
The points \(P\) and \(Q\) are the points of intersection of \(H\) and \(l\)

1. Determine, in terms of \(c\), the coordinates of \(P\) and the coordinates of \(Q\) The point \(R\) is the midpoint of \(P Q\)
2. Show that, as \(C\) varies, the coordinates of \(R\) satisfy the equation $$x y = - \frac { c ^ { 2 } } { a }$$ where \(a\) is a constant to be determined.