Let P be the plane passing through the intersection of the planes
r→.(i+3k−k)=5 and r→ .(2i−j+k)=3,
and the point (2, 1, –2). Let the position vectors of the points X and Y be
i−2j+4k and 5i−j+2k
respectively. Then the points
Let P be the plane passing through the intersection of the planes
r→.(i+3k−k)=5 and r→ .(2i−j+k)=3,
and the point (2, 1, –2). Let the position vectors of the points X and Y be
i−2j+4k and 5i−j+2k
respectively. Then the points
- X and X + Y are on the same side of P
- Y and Y – X are on the opposite sides of P
- X and Y are on the opposite sides of P
- X + Y and X – Y are on the same side of P
The Correct Option is A
Solution and Explanation
The correct option is(C): X and Y are on the opposite sides of P.
Let the equation of required plane
\(\pi:(x+3y-z-5)+λ(2x-y+z-3)=0\)
\(∵(2,1,-2)\,\text{lies on it so,} 2+λ(-2)=0\)
⇒λ=1
Hence,
\(\pi:3x+2y-8=0\)
\(∵\pi{x}=-9,\pi{y},\pi_{x+y}=4\)
\(\pi_{x+y}=-22\,and\,\pi_{y-x}=6\)
Clearly, X and Y are on opposite sides of plane π.
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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.