JEE MAIN - Mathematics (2024 - 5th April Morning Shift - No. 23)
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
108
Explanation
Equation of focal chord
$$y-0=\tan \theta .(x-3)$$
Distance from origin
$$\begin{aligned} & d=\left|\frac{-3 \tan \theta}{\sqrt{1+\tan ^2 \theta}}\right| \\ & I=4 \times 3 \operatorname{cosec}^2 \theta \\ & I. d^2=\frac{9 \tan ^2 \theta}{1+\tan ^2 \theta} \times 12 \operatorname{cosec}^2 \theta \\ & =\frac{108 \operatorname{cosec}^2 \theta}{1+\cot ^2 \theta}=108 \end{aligned}$$
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