Rectangular hyperbola tangent intersection

1. The point \(\mathrm { P } \left( 6 \mathrm { t } , \frac { 6 } { \mathrm { t } } \right) , t \neq 0\), lies on the rectangular hyperbola \(H\) with equation \(x y = 36\) (a) Show that an equation for the tangent to \(H\) at \(P\) is

$$y = - \frac { 1 } { t ^ { 2 } } x + \frac { 12 } { t }$$ The tangent to \(H\) at the point \(A\) and the tangent to \(H\) at the point \(B\) meet at the point \(( - 9,12 )\).
(b) Find the coordinates of \(A\) and \(B\).

7. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\), where \(c\) is a constant. The point \(P \left( c t , \frac { c } { t } \right)\) is a general point on \(H\).

1. Show that the tangent to \(H\) at \(P\) has equation $$t ^ { 2 } y + x = 2 c t$$ The tangents to \(H\) at the points \(A\) and \(B\) meet at the point \(( 15 c , - c )\).
2. Find, in terms of \(c\), the coordinates of \(A\) and \(B\).

10. The point \(P \left( 6 t , \frac { 6 } { t } \right) , t \neq 0\), lies on the rectangular hyperbola \(H\) with equation \(x y = 36\).

1. Show that an equation for the tangent to \(H\) at \(P\) is $$y = - \frac { 1 } { t ^ { 2 } } x + \frac { 12 } { t }$$ The tangent to \(H\) at the point \(A\) and the tangent to \(H\) at the point \(B\) meet at the point \(( - 9,12 )\).
2. Find the coordinates of \(A\) and \(B\).

9. The rectangular hyperbola \(H\) has cartesian equation \(x y = 9\) The points \(P \left( 3 p , \frac { 3 } { p } \right)\) and \(Q \left( 3 q , \frac { 3 } { q } \right)\) lie on \(H\), where \(p \neq \pm q\).

1. Show that the equation of the tangent at \(P\) is \(x + p ^ { 2 } y = 6 p\).
2. Write down the equation of the tangent at \(Q\). The tangent at the point \(P\) and the tangent at the point \(Q\) intersect at \(R\).
3. Find, as single fractions in their simplest form, the coordinates of \(R\) in terms of \(p\) and \(q\).

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 \(H\).

1. Show that the equation of the tangent at point \(P\) is $$p ^ { 2 } y + x = 10 p$$
2. Write down the equation of the tangent at point \(Q\). The tangents at \(P\) and \(Q\) meet at the point \(N\).
   Given \(p + q \neq 0\),
3. 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 \(P Q\).
4. Find the value of \(p ^ { 2 } q ^ { 2 }\).

6. A parabola \(C\) has equation \(y ^ { 2 } = 4 a x , \quad a > 0\) The points \(P \left( a p ^ { 2 } , 2 a p \right)\) and \(Q \left( a q ^ { 2 } , 2 a q \right)\) lie on \(C\), where \(p \neq 0 , q \neq 0 , p \neq q\).

1. Show that an equation of the tangent to the parabola at \(P\) is $$p y - x = a p ^ { 2 }$$
2. Write down the equation of the tangent at \(Q\). The tangent at \(P\) meets the tangent at \(Q\) at the point \(R\).
3. Find, in terms of \(p\) and \(q\), the coordinates of \(R\), giving your answers in their simplest form. Given that \(R\) lies on the directrix of \(C\),
4. find the value of \(p q\).

8. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\), where \(c\) is a positive constant. The point \(P \left( c t , \frac { c } { t } \right) , t \neq 0\), is a general point on \(H\). An equation for the tangent to \(H\) at \(P\) is given by $$y = - \frac { 1 } { t ^ { 2 } } x + \frac { 2 c } { t }$$ The points \(A\) and \(B\) lie on \(H\).
The tangent to \(H\) at \(A\) and the tangent to \(H\) at \(B\) meet at the point \(\left( - \frac { 6 } { 7 } c , \frac { 12 } { 7 } c \right)\).
Find, in terms of \(c\), the coordinates of \(A\) and the coordinates of \(B\).

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

Given that \(P \left( c t , \frac { c } { t } \right) , t \neq 0\), is a general point on \(H\),

1. use calculus to show that the equation of the tangent to \(H\) at \(P\) can be written as $$t ^ { 2 } y + x = 2 c t$$ The points \(A\) and \(B\) lie on \(H\).
   The tangent to \(H\) at \(A\) and the tangent to \(H\) at \(B\) meet at the point \(\left( - \frac { 8 c } { 5 } , \frac { 3 c } { 5 } \right)\).
   Given that the \(x\) coordinate of \(A\) is positive,
2. find, in terms of \(c\), the coordinates of \(A\) and the coordinates of \(B\).

1. A rectangular hyperbola \(H\) has equation \(x y = 25\)

The point \(P \left( 5 t , \frac { 5 } { t } \right) , t \neq 0\), is a general point on \(H\).

1. Show that the equation of the tangent to \(H\) at \(P\) is \(t ^ { 2 } y + x = 10 t\) The distinct points \(Q\) and \(R\) lie on \(H\). The tangent to \(H\) at the point \(Q\) and the tangent to \(H\) at the point \(R\) meet at the point \(( 15 , - 5 )\).
2. Find the coordinates of the points \(Q\) and \(R\).