9. The rectangular hyperbola, \(H\), has cartesian equation \(x y = 25\)
- Show that an equation of the normal to \(H\) at the point \(P \left( 5 p , \frac { 5 } { p } \right) , p \neq 0\), is
$$y - p ^ { 2 } x = \frac { 5 } { p } - 5 p ^ { 3 }$$
This normal meets the line with equation \(y = - x\) at the point \(A\).
- Show that the coordinates of \(A\) are
$$\left( - \frac { 5 } { p } + 5 p , \frac { 5 } { p } - 5 p \right)$$
The point \(M\) is the midpoint of the line segment \(A P\).
Given that \(M\) lies on the positive \(x\)-axis, - find the exact value of the \(x\) coordinate of point \(M\).