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JEE Main 22-Jan-2025 Paper - Shift 2 — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEEJEE Main 22-Jan-2025 Paper - Shift 24 Marks
Question
Let the distance between two parallel lines be 5 units and a point P lie between the lines at a unit distance from one of them. An equilateral triangle PQR is formed such that Q lies on one of the parallel lines, while R lies on the other. Then $(Q R)^2$ is equal to _________.
✓

Answer

(28)
Sol.
Image

$
PR=\operatorname{cosec} \theta, PQ=4 \sec (30+\theta)
$
For equilateral
$
\begin{array}{l}
d=PR=PQ \\
\Rightarrow \cos \left(\theta+30^{\circ}\right)=4 \sin \theta \\
\Rightarrow \frac{\sqrt{3}}{2} \cos \theta-\frac{1}{2} \sin \theta=4 \sin \theta \\
\Rightarrow \tan \theta=\frac{1}{3 \sqrt{3}} \\
QR^2=d^2=\operatorname{cosec}^2 \theta=28
\end{array}
$

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