JEE Mathematics 2024 Question with Solution

Given: A line with direction ratios $2, 1, 2$ meets the lines $x = y + 2 = z$ and $x + 2 = 2y = 2z$ at points $P$ and $Q$ respectively.

Find: The value of $l^2$, where $l$ is the perpendicular distance from $(1, 2, 12)$ to line $PQ$.

Points on the given lines can be written as

and

Since the line $PQ$ has direction ratios $2, 1, 2$, its direction components satisfy

Solving these equations gives $t = 6$ and $s = 2$.

Therefore,

Hence the line $PQ$ is

Let the foot of the perpendicular from $A(1, 2, 12)$ to $PQ$ be

Since $\overrightarrow{AF} \cdot \overrightarrow{PQ} = 0$, solving gives $\lambda = 2$.

So,

Now,

Therefore,

The correct option is B.