Question

Find the length of perpendicular from $P(2,-3,1)$ to the line $\frac{x-1}{2}=\frac{y-3}{3}=\frac{z+2}{-1}$

• A. $5$
• B. $\sqrt{\frac{531}{14}}$
• C. $\sqrt{50}$
• D. $\sqrt{\frac{221}{3}}$

Answer 1

Answer: (B)

$\sqrt{\frac{531}{14}}$

Let take a point on line $(2\lambda +1,3\lambda +3,-\lambda -2)$

Direction ratio's of line which is perpendicular to given line,

$(2\lambda +1-2,3\lambda +3+3,-\lambda -2-1)$

$(2\lambda -1,3\lambda +6,-\lambda -3)$

And the direction ratio's of given line are $2,3,-1$.

These two lines are perpendicular, so

$(2\lambda -1)\cdot 2+(3\lambda +6)\cdot 3+(-\lambda -3)+(-1)=0$

$\lambda =\frac{-19}{14}$

So point is $(\frac{-24}{14},\frac{-15}{14},\frac{-9}{14})$

So distance between $(\frac{-24}{14},\frac{-15}{14},\frac{-9}{14})$ and $(1,3,-2)$ is $\sqrt{\frac{531}{14}}$...................(by distance formula)