Question

The foot of the perpendicular drawn from the point $(1,8,4)$ on the line joining the points $(0,–11,4)$ and $(2,–3,1)$ is

• A. $(4,5,2)$
• B. $(–4,5,2)$
• C. $(4,–5,2)$
• D. $(4,5,–2)$

Answer 1

The correct option is D $(4,5,–2)$

Let $P\equiv (1,8,4)$
Equation of line joining points $(0,–11,4)$ and $(2,–3,1)$
$\frac{x-2}{2}=\frac{y+3}{8}=\frac{z-1}{-3}=\lambda$
Any point on the line is $Q(2\lambda +2,8\lambda -3,1-3\lambda )$
DR's of $PQ(2\lambda +1,8\lambda -11,-3\lambda -3)$
Now $PQ\perp$ to the line
$\therefore (2\lambda +1)2+(8\lambda -11)8+(-3\lambda -3)(-3)=0$
$77\lambda -77=0\Rightarrow \lambda =1$
$Q(4,5,–2)$