JEE MAIN - Mathematics (2024 - 29th January Morning Shift - No. 24)
A line with direction ratios $$2,1,2$$ meets the lines $$x=y+2=z$$ and $$x+2=2 y=2 z$$ respectively at the points $$\mathrm{P}$$ and $$\mathrm{Q}$$. If the length of the perpendicular from the point $$(1,2,12)$$ to the line $$\mathrm{PQ}$$ is $$l$$, then $$l^2$$ is __________.
Answer
65
Explanation
Let $$\mathrm{P}(\mathrm{t}, \mathrm{t}-2, \mathrm{t})$$ and $$\mathrm{Q}(2 \mathrm{~s}-2, \mathrm{~s}, \mathrm{~s})$$
D.R's of PQ are 2, 1, 2
$$\begin{aligned} & \frac{2 \mathrm{~s}-2-\mathrm{t}}{2}=\frac{\mathrm{s}-\mathrm{t}+2}{1}=\frac{\mathrm{s}-\mathrm{t}}{2} \\ & \Rightarrow \mathrm{t}=6 \text { and } \mathrm{s}=2 \\ & \Rightarrow \mathrm{P}(6,4,6) \text { and } \mathrm{Q}(2,2,2) \\ & \mathrm{PQ}: \frac{\mathrm{x}-2}{2}=\frac{\mathrm{y}-2}{1}=\frac{\mathrm{z}-2}{2}=\lambda \end{aligned}$$
Let $$\mathrm{F}(2 \lambda+2, \lambda+2,2 \lambda+2)$$
$$\mathrm{A}(1,2,12)$$
$$\overrightarrow{\mathrm{AF}} \cdot \overrightarrow{\mathrm{PQ}}=0$$
$$\therefore \lambda=2$$
So $$F(6,4,6)$$ and $$A F=\sqrt{65}$$
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