Problem

8. A rectangular hyperbola, $H$, has Cartesian equation $xy = 16$

The point $P\left(4t, \dfrac{4}{t}\right)$, $t \neq 0$, lies on $H$.

(a) Use calculus to show that an equation of the normal to $H$ at $P$ is

$ty - t^3x = 4 - 4t^4$

The point $A$ on $H$ has parameter $t = 2$

The normal to $H$ at $A$ meets $H$ again at the point $B$.

(b) Determine the exact value of the length of $AB$.

The tangent to $H$ at $A$ meets the $y$-axis at the point $C$.

(c) Determine the exact area of triangle $ABC$.