JEE 2023 — Mathematics PYQ

1. Identify the vectors from the line equations

A line equation in the form $\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}$ gives us a point on the line $\vec{a}$ and the direction vector $\vec{b}$.

For Line 1:

• Point $\overrightarrow{a_1}=(5,2,4)$

• Direction $\overrightarrow{b_1}=(1,2,-3)$

For Line 2:

• Point $\overrightarrow{a_2}=(-3,-5,1)$

• Direction $\overrightarrow{b_2}=(1,4,-5)$

2. Find the difference between points ($\overrightarrow{a_2}-\overrightarrow{a_1}$)

3. Calculate the cross product of direction vectors ($\overrightarrow{b_1}\times \overrightarrow{b_2}$)

The magnitude $|\overrightarrow{b_1}\times \overrightarrow{b_2}|=\sqrt{2^2+2^2+2^2}=\sqrt{12}=2\sqrt{3}$.

4. Apply the Shortest Distance Formula

The formula for the shortest distance ($d$) between two skew lines is:

Calculate the dot product:

Final Calculation:

To rationalize, multiply the top and bottom by $\sqrt{3}$:

Correct Option: (3) $6\sqrt{3}$