Question

A perpendicular is drawn from the point $A(1,8,4)$ to the line joining the points $B(0,-1,3)$ and $C(2,-3,-1)$. The co-ordinates of the foot of the perpendicular is:

• A. $(\frac{5}{3},\frac{2}{3},\frac{19}{3})$
• B. $(\frac{5}{3},\frac{2}{3},\frac{-19}{3})$
• C. $(\frac{5}{3},\frac{-2}{3},\frac{19}{3})$
• D. $(\frac{-5}{3},\frac{2}{3},\frac{19}{3})$

Answer 1

The correct option is D $(\frac{-5}{3},\frac{2}{3},\frac{19}{3})$

Equation of line $BC\text{is :}\frac{x}{2}=\frac{y+1}{-2}=\frac{z-3}{-4}$
Any point $D$ on the line can be taken as: $(2t,-2t-1,-4t+3)$
Now, $\overrightarrow{DA}=(1-2t,9+2t,1+4t)$
For foot of perpendicular, $\overrightarrow{DA}.(2i-2j-4k)=0$
$\Rightarrow 2-4t-18-4t-4-16t=0$
$\Rightarrow t=-\frac{5}{6}$
Hence, we have $D=(\frac{-5}{3},\frac{2}{3},\frac{19}{3})$