Question

A line with directiion ratios 2, 7, -5 is intercepted between the lines
$\frac{x-5}{3}=\frac{y-7}{-1}=\frac{z+2}{1}and\frac{x+3}{-3}=\frac{y-3}{2}=\frac{z-6}{4}$. Find the length intercepted between the given lines.___

• A. $\sqrt{78}$
• B. 9
• C. $\sqrt{85}$
• D. 10

Answer 1

The correct option is A

$\sqrt{78}$

The general points on the given lines are respectively
$P(5+3t,7-t,-2+t)andQ(-3-3s,3+2s,6+4s).$
Direction numbers of PQ are
$<-3-3s-5-3t,3+2s-7+t,6+4s+2-t>$
i.e.$<-8-3s-3t,-4+2s+t,8+4s-t>$
If PQ is the desired line then direction numbers of PQ should be proportional to < 2, 7, -5> therefore,
$\frac{-8-3s-3t}{2}=\frac{-4+2s+t}{7}=\frac{8+4s-t}{-5}$
Taking first and second numbers, we get
$-56-21s-21t=-8+4s+2t\Rightarrow 25s+23t=-48\left(i\right)$
taking second and third member, we get
$20-10s-5t=56+28s-7t\Rightarrow 38s-2t=-36\left(ii\right)$
Solving (i) and (ii) for $t$ and $s$ we get
$s=-1$ and $t=-1$. The coordinates of $P$ and $Q$ are respectively
$(5+3(-1),7-(-1),-2-1)=(2,8,-3)$
and $(-3-3(-1),3+2(-1),6+4(-1\left)\right)=(0,1,2)$
$\therefore$ the said line intersects the given lines in the points $(2,8,-3)$ and $(0,1,2)$ respectively.
Length of the line intercepted between the given lines
$=|PQ|\sqrt{(0-2{)}^2+(1-8{)}^2+(2+3{)}^2}=\sqrt{78}.$