Rectangular hyperbola normal equation

6. The rectangular hyperbola, \(H\), has cartesian equation $$x y = 36$$ The three points \(P \left( 6 p , \frac { 6 } { p } \right) , Q \left( 6 q , \frac { 6 } { q } \right)\) and \(R \left( 6 r , \frac { 6 } { r } \right)\), where \(p , q\) and \(r\) are distinct, non-zero values, lie on the hyperbola \(H\).

1. Show that an equation of the line \(P Q\) is $$p q y + x = 6 ( p + q )$$ Given that \(P R\) is perpendicular to \(Q R\),
2. show that the normal to the curve \(H\) at the point \(R\) is parallel to the line \(P Q\).

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.

1. 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 \(R\),
2. find the coordinates of \(Q\) and the coordinates of \(R\), giving your answers in terms of \(p\).
3. Given also that the area of triangle \(O Q R\) is 512 , find the possible values of \(p\).
   

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

The point \(P \left( c t , \frac { c } { t } \right)\), where \(t > 0\), lies on \(H\)

1. Use calculus to show that an equation of the normal to \(H\) at \(P\) is $$t ^ { 3 } x - t y = c \left( t ^ { 4 } - 1 \right)$$ The parabola \(C\) has equation \(y ^ { 2 } = 6 x\)
   The normal to \(H\) at the point with coordinates \(( 8,2 )\) meets \(C\) at the point \(Q\) where \(y > 0\)
2. Determine the exact coordinates of \(Q\) Given that
   • the point \(R\) is the focus of \(C\)
   • the line \(l\) is the directrix of \(C\)
   • the line through \(Q\) and \(R\) meets \(l\) at the point \(S\)
   • determine the exact length of \(Q S\)

9. The rectangular hyperbola, \(H\), has cartesian equation \(x y = 25\)

1. 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 \(A\).
2. 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,
3. find the exact value of the \(x\) coordinate of point \(M\).