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 :
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 :
Updated On: Jul 7, 2022
- $\sqrt{29}$
- $\sqrt{33}$
- $\sqrt{53}$
- $\sqrt{66}$
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The Correct Option is C
Solution and Explanation
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 $\sqrt{1^2 + 6^2 + 4^2} = \sqrt{53}$
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Notes on Plane
Concepts Used:
Plane
A surface comprising all the straight lines that join any two points lying on it is called a plane in geometry. A plane is defined through any of the following uniquely:
- Using three non-collinear points
- Using a point and a line not on that line
- Using two distinct intersecting lines
- Using two separate parallel lines
Properties of a Plane:
- In a three-dimensional space, if there are two different planes than they are either parallel to each other or intersecting in a line.
- A line could be parallel to a plane, intersects the plane at a single point or is existing in the plane.
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