Question

The coordinate of foot of the perpendicular drawn from the point $A(1,0,3)$ to the line joining the points $B(4,7,1)$ and $C(3,5,3)$ is

• A. $(1,2-2)$
• B. $(\frac{5}{3},\frac{7}{3},\frac{17}{3})$
• C. $(\frac{1}{3},\frac{2}{3},\frac{-2}{3})$
• D. $(\frac{-5}{3},\frac{-7}{3},\frac{-17}{3})$

Answer 1

The correct option is B $(\frac{5}{3},\frac{7}{3},\frac{17}{3})$

$A=(1,0,3)$.
D.r' s of the line $BC=(1,2,-2)$
Equation of the line BC is $\frac{x-4}{1}=\frac{y-7}{2}=\frac{z-1}{-2}$
Any point $P$ on the line $BC$ is $(4+t,7+2t,1-2t)$
Let $P$ be the foot of perpendicular
D.r' s of$AP=(3+t,7+2t,-2t-2)$
$AP$ is perpendicular to $BC$
$\Rightarrow 1(3+t)+2(7+2t)-2(-2t-2)=0$
$\Rightarrow t=\frac{-7}{3}P=(4-\frac{7}{3},7-\frac{14}{3},1+\frac{14}{3})$
$\Rightarrow P=(\frac{5}{3},\frac{7}{3},\frac{17}{3})$