7. The rectangular hyperbola, \(H\), has cartesian equation \(x y = 25\)
The point \(P \left( 5 p , \frac { 5 } { p } \right)\), and the point \(Q \left( 5 q , \frac { 5 } { q } \right)\), where \(p , q \neq 0 , p \neq q\), are points on the rectangular hyperbola \(H\).
- Show that the equation of the tangent at point \(P\) is
$$p ^ { 2 } y + x = 10 p$$
- Write down the equation of the tangent at point \(Q\).
The tangents at \(P\) and \(Q\) meet at the point \(N\).
Given \(p + q \neq 0\), - show that point \(N\) has coordinates \(\left( \frac { 10 p q } { p + q } , \frac { 10 } { p + q } \right)\).
The line joining \(N\) to the origin is perpendicular to the line \(P Q\).
- Find the value of \(p ^ { 2 } q ^ { 2 }\).