IAL 2026 Jan FP1 Q9

Question

[!problem]

The rectangular hyperbola $H$ has equation $xy = c^2$ where $c > 0$.

The point $P$ lies on $H$ and has coordinates $\left(ct, \dfrac{c}{t}\right)$ where $t > 0$.

The line $l_1$ is the tangent to $H$ at $P$.

(a) Use calculus to show that an equation for $l_1$ is

$x = t^2y = 2ct$

(3)

The line $l_2$ is the normal to $H$ at $P$.

(b) Determine an equation for $l_2$.

(1)

Given that

• $l_1$ crosses the $x$-axis at the point $X$ and crosses the $y$-axis at the point $Y$
• $l_2$ crosses the $y$-axis at the point $Z$

(c) show that the area of the triangle $XYZ$ is given by

$\dfrac{c^2(1 + t^4)}{2}$

(4)

When $t = 3$

• the area of the triangle $XYZ$ is 410
• the line from $P$ through the origin meets $H$ again at the point $Q$

(d) Determine the exact length of $PQ$, giving your answer as a simplified surd.

(3)