Question

The foot of the perpendicular from the origin to the line joining the points $(9,-4,-5)$ and $(-11,0,1)$ is

• A. $(-2,-1,-2)$
• B. $(-1,-2,-2)$
• C. $(-1,-2,-3)$
• D. $(-2,-2,-1)$

Answer 1

The correct option is B $(-1,-2,-2)$

Equation of line joining $(9,-4,-5)$ and $(-11,0,1)$ is given by,
$\frac{x+11}{-20}=\frac{y}{4}=\frac{z-1}{6}=k$ (say)
So any point on this line is given by, $P(-20k-11,4k,6k+1)$
Let origin is $O(0,0,0)$, so direction ratios of $PO$ are $-20k-11,4k,6k+1$
For $P$ to be foot of perpendicular from origin to this line,
$-20(-20k-11)+4\left(4k\right)+6(6k+1)=0$
$\Rightarrow k=\frac{-1}{2}$
Hence, required point is $P(-1,-2,-2)$