- The points \(P \left( 9 p ^ { 2 } , 18 p \right)\) and \(Q \left( 9 q ^ { 2 } , 18 q \right) , p \neq q\), lie on the parabola \(C\) with equation
$$y ^ { 2 } = 36 x$$
The line \(l\) passes through the points \(P\) and \(Q\)
- Show that an equation for the line \(l\) is
$$( p + q ) y = 2 ( x + 9 p q )$$
The normal to \(C\) at \(P\) and the normal to \(C\) at \(Q\) meet at the point \(A\).
- Show that the coordinates of \(A\) are
$$\left( 9 \left( p ^ { 2 } + q ^ { 2 } + p q + 2 \right) , - 9 p q ( p + q ) \right)$$
Given that the points \(P\) and \(Q\) vary such that \(l\) always passes through the point \(( 12,0 )\)
- find, in the form \(y ^ { 2 } = \mathrm { f } ( x )\), an equation for the locus of \(A\), giving \(\mathrm { f } ( x )\) in simplest form.