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.
Top Questions on Angle between Two Planes
- If the planes $ 2x + 3y + 4z + 7 = 0 $ and $ 4x + ky + 8z + 1 = 0 $ are parallel, then the equation of the plane passing through the point $ (k, k, k) $ and having the direction ratios of its normal as $ (k-1, k, k+1) $ is
- AP EAPCET - 2023
- Mathematics
- Angle between Two Planes
- If the distance between the plane αx-2y+z=k and the plane containing the lines \(\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}\) and \(\frac{x-2}{3}=\frac{y-3}{4}=\frac{z-4}{5}\) is \(\sqrt{6}\), then|k| is :
- WBJEE - 2023
- Mathematics
- Angle between Two Planes
- Let the points on the plane P be equidistant from the points (–4, 2, 1) and (2, –2, 3). Then the acute angle between the plane P and the plane 2x + y + 3z = 1 is
- JEE Main - 2022
- Mathematics
- Angle between Two Planes
- The angle between the planes $3x + 4y + 5z = 3$ and $4 x-3 y + 5z = 9$ is equal to
- KEAM
- Mathematics
- Angle between Two Planes
- Two planes $a_1x + b_1y + c_1z + d_1 = 0$ and $a_2x + b_2y + c_2z + d_2 = 0 $ are parallel if :
- Mathematics
- Angle between Two Planes
Questions Asked in JEE Main exam
Let \( ABC \) be a triangle. Consider four points \( p_1, p_2, p_3, p_4 \) on the side \( AB \), five points \( p_5, p_6, p_7, p_8, p_9 \) on the side \( BC \), and four points \( p_{10}, p_{11}, p_{12}, p_{13} \) on the side \( AC \). None of these points is a vertex of the triangle \( ABC \). Then the total number of pentagons that can be formed by taking all the vertices from the points \( p_1, p_2, \ldots, p_{13} \) is ___________.
- JEE Main - 2026
- Coordinate Geometry
- A voltage regulating circuit consisting of a Zener diode having breakdown voltage of $10\,\text{V}$ and maximum power dissipation of $0.4\,\text{W}$ is operated at $15\,\text{V}$. The approximate value of protective resistance in this circuit is ___ $\Omega$.
- JEE Main - 2026
- Semiconductor electronics: materials, devices and simple circuits
- Two point charges of \(1\,\text{nC}\) and \(2\,\text{nC}\) are placed at two corners of an equilateral triangle of side \(3\) cm. The work done in bringing a charge of \(3\,\text{nC}\) from infinity to the third corner of the triangle is ________ \(\mu\text{J}\). \[ \left(\frac{1}{4\pi\varepsilon_0}=9\times10^9\,\text{N m}^2\text{C}^{-2}\right) \]
- JEE Main - 2026
- Electrostatics
Consider the following two reactions A and B:
The numerical value of [molar mass of $x$ + molar mass of $y$] is ___.
- JEE Main - 2026
- Organic Chemistry
Consider an A.P. $a_1,a_2,\ldots,a_n$; $a_1>0$. If $a_2-a_1=-\dfrac{3}{4}$, $a_n=\dfrac{1}{4}a_1$, and \[ \sum_{i=1}^{n} a_i=\frac{525}{2}, \] then $\sum_{i=1}^{17} a_i$ is equal to
- JEE Main - 2026
- Arithmetic Progression
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.