Verified Solution • 2025–26 Session moderate

QJEEM24Mafe06: JEE Main 2024 Mathematics - Class 11 JEEM24Mafe06 Solution: Suppose is a focal chord of the parabola of length and slope .…

Suppose AB\mathrm{AB} is a focal chord of the parabola y2=12xy^2=12 x of length ll and slope \mathrm{m}<\sqrt{3} . If the distance of the chord AB\mathrm{AB} from the origin is d\mathrm{d} , then l d2l \mathrm{~d}^2 is equal to _________.

Step-by-Step Verified Solution

Equation of focal chord

y−0=tan⁡θ.(x−3)y-0=\tan \theta .(x-3)

Distance from origin

amp;d=∣−3tan⁡θ1+tan⁡2θ∣amp;I=4×3cosec⁡2θamp;I.d2=9tan⁡2θ1+tan⁡2θ×12cosec⁡2θamp;=108cosec⁡2θ1+cot⁡2θ=108\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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