If the plane 2x + y – 5z = 0 is rotated about its line of intersection with the plane 3x – y + 4z – 7 = 0 by an angle of π/2, then the plane after the rotation passes through the point:
- (2, –2, 0)
- (–2, 2, 0)
- (1, 0, 2)
- (–1, 0, –2)
The Correct Option is C
Solution and Explanation
The correct answer is (C) : (1, 0, 2)
\(P1: 2x +y - 52 = 0, P2: 3x - y + 4z-7=0\)
Family of planes P1 and P2
\(P: P1+λP2\)
\(∴ P: (2 + 3λ)x + (1 - λ)y+ (-5 + 4λ)z - 7λ=0 \)
\(∵ P ⊥ P1 ∴ 4 + 6 λ +1 - λ + 25 - 20λ=0\)
\(λ = 2\)
\(P: 8x - y + 32 - 14 = 0\)
It passes through the point \((1, 0, 2).\)
Top Questions on Three Dimensional Geometry
- Find the length of the diagonal of a rectangle with length 6 cm and breadth 8 cm.
- MHT CET - 2025
- Mathematics
- Three Dimensional Geometry
- Find the area of a circle whose radius is 7 cm.
- MHT CET - 2025
- Mathematics
- Three Dimensional Geometry
- The area of a triangle with base 12 cm and height 5 cm is?
- MHT CET - 2025
- Mathematics
- Three Dimensional Geometry
- A straight line drawn from the point P(1,3, 2), parallel to the line \(\frac{x-2}{1}=\frac{y-4}{2}=\frac{z-6}{1}\), intersects the plane L1 : x - y + 3z = 6 at the point Q. Another straight line which passes through Q and is perpendicular to the plane L1 intersects the plane L2 : 2x - y + z = -4 at the point R. Then which of the following statements is (are) TRUE ?
- JEE Advanced - 2024
- Mathematics
- Three Dimensional Geometry
- Let y ∈ R be such that the lines \(L_1:\frac{x+11}{1}=\frac{y+21}{2}=\frac{z+29}{3}\) and \(L_2:\frac{x+16}{3}=\frac{y+11}{2}=\frac{z+4}{\gamma}\) intersect. Let R1 be the point of intersection of L1 and L2. Let O = (0, 0 ,0), and \(\hat{n}\) denote a unit normal vector to the plane containing both the lines L1 and L2.
Match each entry in List-I to the correct entry in List-II.The correct option isList - I List - II (P) γ equals (1) \(-\hat{i}-\hat{j}+\hat{k}\) (Q) A possible choice for \(\hat{n}\) is (2) \(\sqrt{\frac{3}{2}}\) (R) \(\overrightarrow{OR_1}\) equals (3) 1 (S) A possible value of \(\overrightarrow{OR_1}.\hat{n}\) is (4) \(\frac{1}{\sqrt6}\hat{i}-\frac{2}{\sqrt6}\hat{j}+\frac{1}{\sqrt6}\hat{k}\) (5) \(\sqrt{\frac{2}{3}}\) - JEE Advanced - 2024
- Mathematics
- Three Dimensional Geometry
Questions Asked in JEE Main exam
- Consider an A.P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its 11th term is:
- JEE Main - 2025
- Arithmetic Progression
- Let \( f(x) = \frac{x - 5}{x^2 - 3x + 2} \). If the range of \( f(x) \) is \( (-\infty, \alpha) \cup (\beta, \infty) \), then \( \alpha^2 + \beta^2 \) equals to.
- JEE Main - 2025
- Functions
- CrCl\(_3\).xNH\(_3\) can exist as a complex. 0.1 molal aqueous solution of this complex shows a depression in freezing point of 0.558°C. Assuming 100\% ionization of this complex and coordination number of Cr is 6, the complex will be:
- JEE Main - 2025
- Colligative Properties
- Suppose that the number of terms in an A.P. is \( 2k, k \in \mathbb{N} \). If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55, and the last term of the A.P. exceeds the first term by 27, then \( k \) is equal to:
- JEE Main - 2025
- Arithmetic Progression
- If \( \alpha + i\beta \) and \( \gamma + i\delta \) are the roots of the equation \( x^2 - (3-2i)x - (2i-2) = 0 \), \( i = \sqrt{-1} \), then \( \alpha\gamma + \beta\delta \) is equal to:
- JEE Main - 2025
- Complex numbers
Concepts Used:
Three Dimensional Geometry
Mathematically, Geometry is one of the most important topics. The concepts of Geometry are derived w.r.t. the planes. So, Geometry is divided into three major categories based on its dimensions which are one-dimensional geometry, two-dimensional geometry, and three-dimensional geometry.
Direction Cosines and Direction Ratios of Line:
Consider a line L that is passing through the three-dimensional plane. Now, x,y and z are the axes of the plane and α,β, and γ are the three angles the line makes with these axes. These are commonly known as the direction angles of the plane. So, appropriately, we can say that cosα, cosβ, and cosγ are the direction cosines of the given line L.
