JEE Main 2025 Apr 7 Shift 1, Mathematics Q2: Lines in Space

Point on $L_1$: $A=(1,2,3)$ with direction $\overrightarrow{d_1}=(2,3,4)$.

Point on $L_2$: $B=(0,0,5)$ with direction $\overrightarrow{d_2}=(1,\alpha ,1)$.

Vector joining points: $\overrightarrow{AB}=B-A=(-1,-2,2)$.

Compute $\overrightarrow{d_1}\times \overrightarrow{d_2}$:

$\overrightarrow{d_1}\times \overrightarrow{d_2}=\left|\begin{array}{c}\hat{i}\hat{j}\hat{k}\\ 234\\ 1\alpha 1\end{array}\right|=\hat{i}(3-4\alpha )-\hat{j}(2-4)+\hat{k}(2\alpha -3)$

$=(3-4\alpha ,2,2\alpha -3)$

Apply the shortest-distance formula for skew lines:

$\text{S.D.}=\frac{|\overrightarrow{AB}\cdot (\overrightarrow{d_1}\times \overrightarrow{d_2})|}{|\overrightarrow{d_1}\times \overrightarrow{d_2}|}=\frac{5}{\sqrt{6}}$

Numerator dot product:

$\overrightarrow{AB}\cdot (\overrightarrow{d_1}\times \overrightarrow{d_2})=-1(3-4\alpha )+(-2)(2)+2(2\alpha -3)=8\alpha -13$

Squaring both sides of the S.D. equation:

$\frac{(8\alpha -13{)}^2}{(3-4\alpha {)}^2+4+(2\alpha -3{)}^2}=\frac{25}{6}$

Cross-multiplying and simplifying leads to the quadratic $116{\alpha }^2+348\alpha -164=0$.

Sum of roots: By Vieta's, ${\alpha }_1+{\alpha }_2=-\frac{348}{116}=-3$.

$\therefore$ Option (4).