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Question:
The distance between the planes $ 2x-2y+z+3=0 $ and $ 4x-4y+2z+5=0 $ is
The distance between the planes $ 2x-2y+z+3=0 $ and $ 4x-4y+2z+5=0 $ is
Updated On: Jun 23, 2024
- $ 3 $
- $ 6 $
- $ \frac{1}{6} $
- $ \frac{1}{3} $
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The Correct Option is C
Solution and Explanation
Since, the planes $ 2x-2y+z+3=0 $ and $ 2x-2y+z+\frac{5}{2}=0 $ are parallel to each other.
$ \therefore $ Distance between them $ =\frac{|{{c}_{2}}-{{c}_{1}}|}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}} $
$ =\frac{\left| \frac{5}{2}-3 \right|}{\sqrt{4+4+1}} $
$ =\frac{\frac{1}{2}}{3}=\frac{1}{6} $
$ \therefore $ Distance between them $ =\frac{|{{c}_{2}}-{{c}_{1}}|}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}} $
$ =\frac{\left| \frac{5}{2}-3 \right|}{\sqrt{4+4+1}} $
$ =\frac{\frac{1}{2}}{3}=\frac{1}{6} $
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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.
- If there are two different lines that are perpendicular to the same plane then they must be parallel to each other.
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