JEE Main 2024 — 8 April, Shift 2 — Question 22

$\frac{\left\lvert\, \mathrm{A}\left(\frac{17}{14}, \frac{48}{14}, \frac{79}{14}\right)\right.}{\frac{\mathrm{x}}{1}=\frac{\mathrm{y}-1}{2}=\frac{\mathrm{z}-2}{3}}$

$\mathrm{P}(\alpha, \beta, \gamma), \quad \overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$

$A(t, 2 t+1,3 t+2)$

$\overrightarrow{\mathrm{QA}}=(\mathrm{t}-1) \hat{\mathrm{i}}+(2 \mathrm{t}-5) \hat{\mathrm{j}}+(3 \mathrm{t}-2) \hat{\mathrm{k}}$

$\overrightarrow{\mathrm{QA}} \cdot \vec{b}=0$

$(\mathrm{t}-1)+2(2 \mathrm{t}-5)+3(3 \mathrm{t}-2)=0$

$14 \mathrm{t}=17$

$\alpha=\frac{20}{14}, \quad \beta=\frac{12}{14} ,\quad \gamma=\frac{102}{14}$

$2 \alpha+\beta+\gamma=\frac{154}{14}=11$