JEE Main 2025 — 4 April, Morning Shift — Question 36

$L_{1}: \frac{x-3}{3}=\frac{y-\alpha}{-1}=\frac{z-3}{1}$

$$L_{2}: \frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-\beta}{4}$$

$a_{1}:(3, \alpha, 3) \quad a^{2}=(-3,-7, \beta)$

$\vec{b}_{1}=3 \hat{i}-\hat{j}+\hat{k} \quad \vec{b}_{2}=-3 \hat{i}+2 \hat{j}+4 \hat{k}$

$d=\frac{\left|\left(\vec{a}_{2}-\vec{a}_{1}\right) \cdot\left(\vec{b}_{1} \times \vec{b}_{2}\right)\right|}{\left|\vec{b}_{1} \times \vec{b}_{2}\right|}=3 \sqrt{30}$

$=\frac{\left|\begin{array}{ccc}6 & \alpha+7 & 3-\beta\\ 3 & -1 & 1\\ -3 & 2 & 4\end{array}\right|}{\sqrt{6^{2}+15^{2}+3^{2}}}=3 \sqrt{30}$

$\Rightarrow|-15 \alpha-3 \beta-132|=270$

$$|5 \alpha+\beta+44|=90$$

$\Rightarrow 5 \alpha+\beta=90-44=46$