Thank you for reporting, we will resolve it shortly
Q.
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
AMUAMU 2015Three Dimensional Geometry
Solution:
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 $P Q$ 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, $P O$ 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 $(3,5,9)$.
$\therefore P O=\sqrt{(3-1)^{2}+(5-2)^{2}+(9-3)^{2}}=7$ units
