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Q.
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 :
Three Dimensional Geometry
Solution:
Let feet of perpendicular is
$(2 \alpha ,3 \alpha + 2, 4 \alpha + 3)$
$\Rightarrow $ Direction ratio of the $\bot$ 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 $\bot$ is (2, 5, 7)
$\Rightarrow $ Length $\bot$ is $\sqrt{1^2 + 6^2 + 4^2} = \sqrt{53}$