JEE Main 2025 — 7 April, Morning Shift — Question 36

$d=\left|\frac{(\vec{a}-\vec{b}) \cdot \vec{p}_{1} \times \vec{p}_{2}}{\left|\vec{p}_{1} \times \vec{p}_{2}\right|}\right|=\frac{5}{\sqrt{6}}$

$\vec{a}-\vec{b}=\hat{i}+2 \hat{j}-2 \hat{k}$

$\overrightarrow{p_{1}} \times \overrightarrow{p_{2}}=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k}\\ 2 & 3 & 4\\ 1 & \alpha & 1\end{array}\right|$

$$=\hat{i}(3-4 \alpha)-\hat{j}(-2)+\hat{k}(2 \alpha-3)$$

$(\vec{a}-\vec{b}) \cdot\left|\overrightarrow{p_{1}} \times \overrightarrow{p_{2}}\right|=3-4 \alpha+4-4 \alpha+6$

$=13-8 \alpha$

$\left|\frac{13-8 \alpha}{\sqrt{(3-4 \alpha)^{2}+4+(2 \alpha-3)^{2}}}\right|=\frac{5}{\sqrt{6}}$

$\left|\frac{13-8 \alpha}{\sqrt{20 \alpha^{2}-36 \alpha+22}}\right|=\frac{5}{\sqrt{6}}$

$=6(13-8 \alpha)^{2}=25\left(20 \alpha^{2}-36 \alpha+22\right)$

$=116 \alpha^{2}+348 \alpha-464=0$

Sum of roots $=-3$