JEE Main 2024 — 4 April, Shift 2 — Question 29

DR's of Line $\mathrm{L} \equiv-1: 1: 2$

DR's of $A B \equiv \alpha-2: \beta-2: \gamma-2$

$\mathrm{AB} \perp_{\mathrm{ar}} \mathrm{L} \Rightarrow 2-\alpha+\beta-2+2 \gamma-4=0$

$2 \gamma+\beta-\alpha=4$

Let $C$ is mid-point of $A B$

$C\left(\frac{\alpha+2}{2}, \frac{\beta+2}{2}, \frac{\gamma+2}{2}\right)$

DR's of $\mathrm{PC}=\frac{\alpha}{2}: \frac{\beta-2}{2}: \frac{\gamma}{2}$

line $L \| P C \Rightarrow \frac{-\alpha}{2}=\frac{\beta-2}{2}=\frac{\gamma}{4}=K$

(let) $\alpha=-2 \mathrm{~K}$

$\beta=2 \mathrm{~K}+2$

$\gamma=4 \mathrm{~K}$ use in $(1) \Rightarrow K=\frac{1}{6}$

value of $\alpha+\beta+6 \gamma=24 \mathrm{~K}+2=6$