JEE Main 2024 — 29 January, Shift 1 — Question 29

Let $P(t, t-2, t)$ and $Q(2 s-2, s, s)$ D.R's of PQ are $2, 1, 2$

$\frac{2 \mathrm{~s}-2-\mathrm{t}}{2}=\frac{\mathrm{s}-\mathrm{t}+2}{1}=\frac{\mathrm{s}-\mathrm{t}}{2}$

$\Rightarrow \mathrm{t}=6$ and $\mathrm{s}=2$

$\Rightarrow \mathrm{P}(6,4,6)$ and $\mathrm{Q}(2,2,2)$

$\mathrm{PQ}: \frac{\mathrm{x}-2}{2}=\frac{\mathrm{y}-2}{1}=\frac{\mathrm{z}-2}{2}=\lambda$

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 $A F=\sqrt{65}$