Parabola
Focal Chord Length — Distance from Origin
nta_pyq_2024_apr
Grade None
Question:
Let the length of the focal chord $PQ$ of the parabola $y^2=12x$ be 15 units. If the distance of $PQ$ from the origin is $p$, then $10p^2$ is equal to
Step-by-Step Solution
Key Concept: For $y^2=12x$, $a=3$. Length of focal chord $=4a\csc^2\theta=12\csc^2\theta=15\Rightarrow\sin^2\theta=4/5$, $\tan\theta=2$. Equation of focal chord through focus $(3,0)$ with slope $m=\tan\theta=2$: $y=2(x-3)$, i.e. $2x-y-6=0$.
$p=6/\sqrt{5}$. $10p^2=72$.
Correct Answer: 72