A line with direction ratios 2,1,2 meets the lines x=y+2=z and x+2=2 y=2 z respectively at…

Question

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 $$\mathrm{PQ}$$ ...

Answer: 65

Accepted Answer

Let $P(t,t-2,t)$ and $Q(2s-2,s,s)$

D.R's of PQ are 2, 1, 2

$\begin{gathered} \frac{2s-2-t}{2}=\frac{s-t+2}{1}=\frac{s-t}{2} \ \Rightarrow t=6s=2 \ \Rightarrow P(6,4,6)Q(2,2,2) \ PQ:\frac{x-2}{2}=\frac{y-2}{1}=\frac{z-2}{2}=\lambda \end{gathered}$

Let $F(2\lambda +2,\lambda +2,2\lambda +2)$

$A(1,2,12)$

$\overset{	o \leftarrow }{AF}\cdot \overset{	o \leftarrow }{PQ}=0$

$	herefore \lambda =2$

So $F(6,4,6)$ and $AF=\sqrt{65}$

Solutions · 1