Question

The foot of the perpendicular drawn from a point with position vector $i+4k$ on the line joining the points having position vector as $-11i+3k$ and $2i-3j+k$ has the position vector

• A. $4i+5j+5k$
• B. $4i+5j-5k$
• C. $5i+4j-5k$
• D. none of these

Answer 1

Answer: (D)

none of these

Equation of line joining $(-11,0,3)$ and $(2,-3,1)$ is given by,

$\frac{x+11}{-13}=\frac{y}{3}=\frac{z-3}{2}=k$ (say)
So any point on this line is given by, $P(-13k-11,3k,2k+3)$
Let given point is $Q(1,0,4)$ so direction ratios of $PO$ are $-13k-12,3k,2k-1$
For $P$ to be foot of perpendicular from origin to this line,
$-13(-13k-12)+3\left(3k\right)+2(2k-1)=0$
$\Rightarrow k=\frac{-77}{91}$
Therefore, required point is $P(0,\frac{-33}{13},\frac{17}{13})$
Hence, option 'D' is correct.