Three Dimensional Geometry Question 13
Question 13 - 2024 (29 Jan Shift 1)
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 point $P$ and $Q$. if the length of the perpendicular from the point $(1,2,12)$ to the line PQ is $l$, then $l^{2}$ is
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Answer (65)
Solution
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
$\frac{2 \mathrm{~s}-2-\mathrm{t}}{2}=\frac{\mathrm{s}-\mathrm{t}+2}{1}=\frac{\mathrm{s}-\mathrm{t}}{2}$
$\Rightarrow \mathrm{t}=6$ 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 $\mathrm{F}(6,4,6)$ and $\mathrm{AF}=\sqrt{65}$
