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
Show Answer
Answer (65)
Solution
Let $P(t, t-2, t)$ and $Q(2 s-2, s, s)$
D.R’s of PQ are 2, 1, 2
$\frac{2 s-2-t}{2}=\frac{s-t+2}{1}=\frac{s-t}{2}$
$\Rightarrow t=6$ and $s=2$
$\Rightarrow P(6,4,6)$ and $Q(2,2,2)$
$PQ: \frac{x-2}{2}=\frac{y-2}{1}=\frac{z-2}{2}=\lambda$
Let $F(2 \lambda+2, \lambda+2,2 \lambda+2)$
$A(1,2,12)$
$\overrightarrow{AF} \cdot \overrightarrow{PQ}=0$
$\therefore \lambda=2$
So $F(6,4,6)$ and $AF=\sqrt{65}$
