Question:
The equation of the plane passing through the mid point of the line of join of the points $(1, 2, 3)$ and $(3, 4, 5) $ and perpendicular to it is
The equation of the plane passing through the mid point of the line of join of the points $(1, 2, 3)$ and $(3, 4, 5) $ and perpendicular to it is
Updated On: Sep 3, 2024
- $ x+y+z=9 $
- $ x+y+z=-9 $
- $ 2x+3y+4z=9 $
- $ 2x+3y+4z=-9 $
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The Correct Option is A
Solution and Explanation
The DR's of the joining of the points $(1,2,3)$ and $(3,4,5)$ are $(3-1,4-2,5-3)$, i.e., $(2,2,2)$
Also, the mid point of the join of the points $(1,2,3)$ and $(3,4,5)$ is $(2,3,4)$
$\therefore$ Equation of plane which passes through $(2,3,4)$ and the D.R.'s of its normal are $(2,2)$ is
$ 2(x-2)+2(y-3)+2(z-4)=0 $
$\Rightarrow x+y+z-9=0$
$\Rightarrow x+y+z=9 .$
Note : If any line is perpendicular to the plane then the DR's of a line is equal to the DR's of the normal to the plane.
Also, the mid point of the join of the points $(1,2,3)$ and $(3,4,5)$ is $(2,3,4)$
$\therefore$ Equation of plane which passes through $(2,3,4)$ and the D.R.'s of its normal are $(2,2)$ is
$ 2(x-2)+2(y-3)+2(z-4)=0 $
$\Rightarrow x+y+z-9=0$
$\Rightarrow x+y+z=9 .$
Note : If any line is perpendicular to the plane then the DR's of a line is equal to the DR's of the normal to the plane.
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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.
- If there are two separate planes which are perpendicular to the same line then they must be parallel to each other.