10. The rectangular hyperbola \(H\) has equation \(x y = 144\). The point \(P\), on \(H\), has coordinates \(\left( 12 p , \frac { 12 } { p } \right)\), where \(p\) is a non-zero constant.
- Show, by using calculus, that the normal to \(H\) at the point \(P\) has equation
$$y = p ^ { 2 } x + \frac { 12 } { p } - 12 p ^ { 3 }$$
Given that the normal through \(P\) crosses the positive \(x\)-axis at the point \(Q\) and the negative \(y\)-axis at the point \(R\),
- find the coordinates of \(Q\) and the coordinates of \(R\), giving your answers in terms of \(p\).
- Given also that the area of triangle \(O Q R\) is 512 , find the possible values of \(p\).
| VIUV SIHI NI JIIIM ION OC | VI4V SIHI NI JINM IONOO | VJYV SIHI NI GLIYM LON OO |