Question

The length of the perpendicular drawn from the point (3, -1, 11) to the line $\frac{x}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ is :

• A. $\sqrt{29}$
• B. $\sqrt{33}$
• C. $\sqrt{53}$
• D. $\sqrt{66}$

Answer 1

Answer: (C)

Let feet of perpendicular is
$(2\alpha ,3\alpha +2,4\alpha +3)$
$\Rightarrow$ Direction ratio of the $\perp$ line is
$2\alpha -3,3\alpha +3,4\alpha -8$. and
Direction ratio of the line 2, 3, 4 are
$\Rightarrow 2(2\alpha -3)+3(3\alpha +3)+4(4\alpha -8)=0$
$\Rightarrow 29\alpha -29=0$
$\Rightarrow \alpha =1$
$\Rightarrow$ Feet of $\perp$ is (2, 5, 7)
$\Rightarrow$ Length $\perp$ is $\sqrt{1^2+6^2+4^2}=\sqrt{53}$