Problem

6. The curve $H$ has equation

$xy = a^2 \quad x > 0$

where $a$ is a positive constant.

The line with equation $y = kx$, where $k$ is a positive constant, intersects $H$ at the point $P$

(a) Use calculus to determine, in terms of $a$ and $k$, an equation for the tangent to $H$ at $P$

The tangent to $H$ at $P$ meets the $x$-axis at the point $A$ and meets the $y$-axis at the point $B$

(b) Determine the coordinates of $A$ and the coordinates of $B$, giving your answers in terms of $a$ and $k$

(c) Hence show that the area of triangle $AOB$, where $O$ is the origin, is independent of $k$