Question

The coordinates of the foot of the perpendicular drawn from the point $A(1,0,3)$ to the join of the points $B(4,7,1)$ and $C(3,5,3)$ are

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

Answer 1

Answer: (A)

$(\frac{5}{3},\frac{7}{3},\frac{17}{3})$

The given point is $P(1,0,3)$ and equation of line passing through $(4,7,1)$ and $(3,5,3)$ is given by,
$\frac{x-4}{1}=\frac{y-7}{2}=\frac{z-1}{-2}=k$(say) .....(1)
So any point on this line is given as$Q(k+4,2k+7,-2k+1)$
Now direction ratios of PQ are,$k+3,2k+7,-2k-2$
Also $PQ\perp \left(1\right)$
$\Rightarrow 1(k+3)+2(2k+7)-2(-2k-2)=0$
$\Rightarrow 9k+21=0\Rightarrow k=\frac{-7}{3}$
Hence, foot of perpendicular drawn on the given line is $(\frac{5}{3},\frac{7}{3},\frac{17}{3})$