IAL 2026 Jan A FP1 Q8

Question

[!problem]

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

The parabola $C$ has equation $y^2 = 4ax$, where $a$ is a positive constant.

The point $P(ap^2, 2ap)$ lies on the parabola $C$.

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

$py = x + ap^2$

(4)

The tangent to $C$ at the point $P$ intersects the directrix of $C$ at the point $B$ and intersects the $x$-axis at the point $D$.

Given that the $y$-coordinate of $B$ is $\dfrac{a}{6}$ and $p > 0$

(b) find, in terms of $a$, the $x$-coordinate of $D$.

(6)

Given that $O$ is the origin,

(c) find, in terms of $a$, the area of the triangle $OPD$, giving your answer in its simplest form.

(2)