Suppose $\mathrm{AB}$ is a focal chord of the parabola $y^2=12 x$ of length $l$ and slope $\mathrm{m}<\sqrt{3}$. If the distance of the chord $\mathrm{AB}$ from the origin is $\mathrm{d}$, then $l \mathrm{~d}^2$ is equal to _________.
Answer (integer)
108
Solution
<p>Equation of focal chord</p>
<p>$y-0=\tan \theta .(x-3)$</p>
<p>Distance from origin</p>
<p>$$\begin{aligned}
& d=\left|\frac{-3 \tan \theta}{\sqrt{1+\tan ^2 \theta}}\right| \\
& I=4 \times 3 \operatorname{cosec}^2 \theta \\
& I. d^2=\frac{9 \tan ^2 \theta}{1+\tan ^2 \theta} \times 12 \operatorname{cosec}^2 \theta \\
& =\frac{108 \operatorname{cosec}^2 \theta}{1+\cot ^2 \theta}=108
\end{aligned}$$</p>
About this question
Subject: Mathematics · Chapter: Conic Sections · Topic: Parabola
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