Let the vertices Q and R of the triangle P Q R lie on the line x+3/5=y-1/2=z+4/3, Q R=5 and the coordinates of the point P be (0,2,3) . If the area of the triangle P Q R is m/n then:

Mathematics · 3D Geometry

JEE Main 2025 — 2 April, Morning Shift — Question 36

Let the vertices $Q$ and $R$ of the triangle $P Q R$ lie on the line $\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}, Q R=5$ and the

coordinates of the point $P$ be $(0,2,3)$ . If the area of the triangle $P Q R$ is $\frac{m}{n}$ then:

$H:(5 \lambda-3,2 \lambda+1,3 \lambda-4)$

< DR of PH>

$<5 \lambda-3,2 \lambda-1,3 \lambda-7>$

$\overrightarrow{P H} \cdot \overrightarrow{Q R}=0$

$\Rightarrow(5 \lambda-3) 5+(2 \lambda-1) 2+(3 \lambda-7) 3=0$

$\Rightarrow 25 \lambda-15+4 \lambda-2+9 \lambda-21=0$

$\Rightarrow 38 \lambda=38$

$\Rightarrow \lambda=1$

$H(2,3,-1)$

$P H=\sqrt{4+1+16}=\sqrt{21}$

$\therefore \quad$ Area $=\frac{1}{2} \times P H \quad Q R$

=\frac{1}{2} \times \sqrt{21} \times 5=\frac{5 \sqrt{21}}{2}=\frac{m}{n}

$2 m-5 \sqrt{21} n=0$

Answer key and solution verified before publishing.

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