Find the coordinates of the foot of the perpendicular drawn from the point $P(0, 2, 3)$ to the line $\frac{x+3}{5} = \frac{y-1}{2} = \frac{z+4}{3}$.

$(2, 3, -1)$

The correct answer is Option (1) → $(2, 3, -1)$ ##

Given point is $P(0, 2, 3)$

Given line is: $\frac{x+3}{5} = \frac{y-1}{2} = \frac{z+4}{3} = \lambda$

General point on the line is $[(5\lambda - 3), (2\lambda + 1), (3\lambda - 4)]$

Direction ratio of the perpendicular line

$[(5\lambda - 3), (2\lambda - 1), (3\lambda - 7)]$

$∴5(5\lambda - 3) + 2(2\lambda - 1) + 3(3\lambda - 7) = 0$

$25\lambda - 15 + 4\lambda - 2 + 9\lambda - 21 = 0$

$38\lambda - 38 = 1$

$\lambda = 1$

$∴$ Foot of perpendicular line is $[(5-3), (2+1), (3-4)] = (2, 3, -1)$