Question

The length of the perpendicular drawn from $(1,2,3)$ to the line $\frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2}$ is

• A. 4 units
• B. 5 units
• C. 6 units
• D. 7 units

Answer 1

Answer: (D)

Let $Q$ be the foot of the perpendicular drawn from the point $P(1,2,3)$ to the given line.

Let the coordinates of $Q$ be
$(3\lambda +6,2\lambda +7,-2\lambda +7)$
$\therefore$ Direction ratios of $PQ$ are proportional to
$3\lambda +6-1,2\lambda +7-2,-2\lambda +7-3$
i.e $3\lambda +5,2\lambda +5,-2\lambda +4$
Direction ratios of the given line are proportional to $3,2,-2$.
Since, $PO$ is perpendicular to the given line.
$\therefore 3(3\lambda +5)+2(2\lambda +5)+(-2)(-2\lambda +4)=0$
$\Rightarrow \lambda =-1$
On putting $\lambda =-1$ in Eq. (i), we obtain the coordinates of $Q$ as $(3,5,9)$.
$\therefore PO=\sqrt{(3-1{)}^2+(5-2{)}^2+(9-3{)}^2}=7$ units