Parabola
Focal Chord — Length Times Distance Squared
nta_pyq_2024_apr
Grade 11
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
Step-by-Step Solution
Key Concept: For focal chord of $y^2=4ax$ ($a=3$): $l=4a\csc^2\theta=12\csc^2\theta$ and $d=3\sin\theta$ (perpendicular distance from origin to the focal chord through $(3,0)$ with inclination $\theta$... actually $d=3|\sin\theta|/\sqrt{1}$... from solution $\ell=12/d^2\cdot9$, so $\ell d^2=108$).
$\ell\cdot d^2=108$.
Correct Answer: 108