Question

Suppose AB is a focal chord of the parabola $y^2 = 12x$ of length $l$ and slope $m<\sqrt{3}$. If the distance of the chord AB from the origin is $d$, then $ld^2$ is equal to _________.

Answer 1

For a focal chord of the parabola $y^2 = 12x$, we have:

$l = 4a \csc^2 \theta$

Given that $l = 12x$ and using the property of a focal chord, we find:

$l = 12 \times \frac{9}{d^2}$

Thus:

$l d^2 = 108$