Question:
A plane passes through $ (1 ,-2 ,1 )$ and is perpendicular to two planes $2x - 2y + z = 0$ and $x - y + 2z = 4, $ then the distance of the plane from the point $ (1, 2, 2)$ is
A plane passes through $ (1 ,-2 ,1 )$ and is perpendicular to two planes $2x - 2y + z = 0$ and $x - y + 2z = 4, $ then the distance of the plane from the point $ (1, 2, 2)$ is
Updated On: Jun 14, 2022
- 0
- 1
- $\sqrt 2$
- $2 \sqrt 2$
Hide Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Let the equation of plane be
$\, \, \, \, \, a\, (x-1) +b\, (y+2)+c\, (z-1) = 0$
which is perpendicular to $ 2 x- 2y + z = 0$ and
$ x - y + 2z = 4.$
$\Rightarrow$ $\hspace10mm 2a- 2b +c = 0\, \, and\, \, a - b + 2c = 0 $
$\Rightarrow$ $\hspace15mm \frac{a}{-3} = \frac{b}{-3} = \frac{c}{0} $
$\Rightarrow$ $\hspace20mm \frac{a}{1} = \frac{b}{1} = \frac{c}{0}.$
So, the equation of plane i s $ x - l + y + 2 = 0 $
or $\hspace70mm x + y + 1 = 0 $
Its distance from the point (1, 2, 2) is
$\hspace50mm \frac{|\, 1\, +\, 2\, +\, 1\, |}{ \sqrt2}= 2 \sqrt 2$
$\, \, \, \, \, a\, (x-1) +b\, (y+2)+c\, (z-1) = 0$
which is perpendicular to $ 2 x- 2y + z = 0$ and
$ x - y + 2z = 4.$
$\Rightarrow$ $\hspace10mm 2a- 2b +c = 0\, \, and\, \, a - b + 2c = 0 $
$\Rightarrow$ $\hspace15mm \frac{a}{-3} = \frac{b}{-3} = \frac{c}{0} $
$\Rightarrow$ $\hspace20mm \frac{a}{1} = \frac{b}{1} = \frac{c}{0}.$
So, the equation of plane i s $ x - l + y + 2 = 0 $
or $\hspace70mm x + y + 1 = 0 $
Its distance from the point (1, 2, 2) is
$\hspace50mm \frac{|\, 1\, +\, 2\, +\, 1\, |}{ \sqrt2}= 2 \sqrt 2$
Was this answer helpful?
0
0
Top Questions on introduction to three dimensional geometry
- Let
\[ f(x) = \begin{cases} x\left( \frac{\pi}{2} + x \right), & \text{if } x \geq 0 \\ x\left( \frac{\pi}{2} - x \right), & \text{if } x < 0 \end{cases} \]
Then \( f'(-4) \) is equal to:- KEAM - 2024
- Mathematics
- introduction to three dimensional geometry
If \( f'(x) = 4x\cos^2(x) \sin\left(\frac{x}{4}\right) \), then \( \lim_{x \to 0} \frac{f(\pi + x) - f(\pi)}{x} \) is equal to:
- KEAM - 2024
- Mathematics
- introduction to three dimensional geometry
- Let \[ f(x) = \frac{|5 - x|(x + 5)}{\tan(x - 5)} \quad {for} \quad x \neq 5. \] Then \[ \lim_{x \to 5} f(x) { is equal to:} \]
- KEAM - 2024
- Mathematics
- introduction to three dimensional geometry
Let \( f(x) = x \sin(x^4) \). Then \( f'(x) \) at \( x = \sqrt[4]{\pi} \) is equal to:
- KEAM - 2024
- Mathematics
- introduction to three dimensional geometry
- Let \( \vec{a}, \vec{b}, \vec{c} \) be three vectors. The angle between \( \vec{a} \) and \( \vec{b} \) is \( 30^\circ \), the angle between \( \vec{a} \) and \( \vec{b} + \vec{c} \) is \( 45^\circ \). If \( |\vec{b}| = \sqrt{6} \) and \( |\vec{c}| = 2\sqrt{2} \), then \( |\vec{b} + \vec{c}| \) is:
- KEAM - 2024
- Mathematics
- introduction to three dimensional geometry
View More Questions
Questions Asked in JEE Advanced exam
- Let \(S=\left\{\begin{pmatrix} 0 & 1 & c \\ 1 & a & d\\ 1 & b & e \end{pmatrix}:a,b,c,d,e\in\left\{0,1\right\}\ \text{and} |A|\in \left\{-1,1\right\}\right\}\), where |A| denotes the determinant of A. Then the number of elements in S is _______.
- JEE Advanced - 2024
- Matrices
- A positive, singly ionized atom of mass number \( A_M \) is accelerated from rest by the voltage \( 192 \, \text{V} \). Thereafter, it enters a rectangular region of width \( w \) with magnetic field \( \vec{B}_0 = 0.1\hat{k} \, \text{T} \). The ion finally hits a detector at the distance \( x \) below its starting trajectory. Which of the following option(s) is(are) correct? \[ \text{(Given: Mass of neutron/proton = } \frac{5}{3} \times 10^{-27} \, \text{kg, charge of the electron = } 1.6 \times 10^{-19} \, \text{C).} \]
- JEE Advanced - 2024
- Electromagnetic Field (EMF)
- A thin stiff insulated metal wire is bent into a circular loop with its two ends extending tangentially from the same point of the loop. The wire loop has mass \( m \) and radius \( r \) and is in a uniform vertical magnetic field \( B_0 \). When a current \( I \) is passed through the loop, the loop turns about the line PQ by an angle \( \theta \). The angle \( \theta \) is given by:
- JEE Advanced - 2024
- Electromagnetic Field (EMF)
- A region in the form of an equilateral triangle (in x-y plane) of height \( L \) has a uniform magnetic field \( \mathbf{B} \) pointing in the \( +z \)-direction. A conducting loop PQR, in the form of an equilateral triangle of the same height \( L \), is placed in the x-y plane with its vertex \( P \) at \( x = 0 \) in the orientation shown in the figure. At \( t = 0 \), the loop starts entering the region of the magnetic field with a uniform velocity \( \mathbf{v} \) along the \( +x \)-direction. The plane of the loop and its orientation remain unchanged throughout its motion.
Which of the following graphs best depicts the variation of the induced emf (\( \mathcal{E} \)) in the loop as a function of the distance (\( x \)) starting from \( x = 0 \)?- JEE Advanced - 2024
- Radioactivity
- A table tennis ball has radius \( \frac{3}{2} \times 10^{-2} \, \text{m} \) and mass \( \frac{22}{7} \times 10^{-3} \, \text{kg} \). It is slowly pushed down into a swimming pool to a depth \( d = 0.7 \, \text{m} \) below the water surface and then released from rest. It emerges from the water surface at speed \( v \), without getting wet, and rises to a height \( H \). Which of the following option(s) is(are) correct?\(\text{(Given: } \pi = \frac{22}{7}, g = 10 \, \text{m/s}^2, \text{density of water} = 1 \times 10^3 \, \text{kg/m}^3, \text{viscosity of water} = 1 \times 10^{-3} \, \text{Pa.s).}\)
- JEE Advanced - 2024
- Radioactivity
View More Questions
Concepts Used:
Introduction to Three Dimensional Geometry
In mathematics, Geometry is one of the most important topics. The concepts of Geometry are defined with respect to the planes. So, Geometry is divided into three categories based on its dimensions which are one-dimensional geometry, two-dimensional geometry, and three-dimensional geometry.
Direction Cosines and Direction Ratios of Line
Let's consider line ‘L’ 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 making with these axes. These are called the plane's direction angles. So, correspondingly, we can very well say that cosα, cosβ, and cosγ are the direction cosines of the given line L.
Read More: Introduction to Three-Dimensional Geometry