JEE Main 2025 — 3 April, Morning Shift — Question 21

$\mathrm{L}_{1}=\frac{\mathrm{x}-1}{2}=\frac{\mathrm{y}-2}{3}=\frac{\mathrm{z}-3}{4}=\mathrm{p}$

$\mathrm{L}_{2}=\frac{\mathrm{x}-6}{1}=\frac{\mathrm{y}}{1}=\frac{\mathrm{z}-4}{-1}=\mathrm{q}$

$\mathrm{A}(2 \mathrm{p}+1,3 \mathrm{p}+2,4 \mathrm{p}+3)$

$B(q+6, q, 4-q)$

D.R. of $\mathrm{PA}=2 \mathrm{p}-3,3 \mathrm{P}+1,4 \mathrm{p}+3$

D.R. of $\mathrm{PB}=\mathrm{q}+2, \mathrm{q}-1,4-\mathrm{q}$

$\frac{2 p-3}{q+2}=\frac{3 p+1}{q-1}=\frac{4 p+3}{4-q}$

$2 p q-2 p-3 q+3=3 p q+6 p+q+2$

$$\begin{gathered} p q+r p+4 q-1=0 \end{gathered}$$

$12 p-3 p q+4-q=4 p q+3 q-4 p-3$

$$\begin{gathered} 7 p q-16 p+4 q-7=0 \end{gathered}$$

$8 p-2 p q-12+3 q=4 p q+8 p+3 q+6$

$6 p q=-18 \quad \therefore p q=-3$

$8 \mathrm{p}+4 \mathrm{q}=4 \quad \Rightarrow 2 \mathrm{p}+\mathrm{q}=1$

$-21-16 p+4 q-7 \quad \Rightarrow 4 p-q=-7$

$16 \mathrm{p}-4 \mathrm{q}=-28 \quad \therefore \mathrm{p}=-1, \mathrm{q}=3$

$\mathrm{A}(-1,-1,-1) \quad \mathrm{B}(9,3,1)$

$\left|\begin{array}{ccc}1 & 0 & 1\\ -1 & -1 & -1\\ 9 & 3 & 1\end{array}\right|=\left|\begin{array}{ccc}0 & -1 & 0\\ -1 & -1 & -1\\ 9 & 3 & 1\end{array}\right|=1(-1+9)=8$