RD Sharma Class 12 Exercise FBQ The Plane Solutions Maths

RD Sharma Class 12 Exercise FBQ The Plane Solutions Maths - Download PDF Free Online

Updated on Jan 25, 2022 11:52 AM IST

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RD Sharma Class 12 Solutions Chapter28 FBQ The Plane - Other Exercise
The Plane Exercise: FBQ
RD Sharma Chapter-wise Solutions

RD Sharma Class 12 Solutions Chapter28 FBQ The Plane - Other Exercise

The Plane Exercise: FBQ

The Plane exercise fill in the blank question 1

Answer:
10
Hint:
Use vector dot product
Given:
x + 2y - 2z = d, d>0
Solution:
Perpendicular distance of the plane ax +by + cz + d = 0

(x, y, z) is d = | \frac{a x_{1} + b y_{1} + c z_{1} + d}{a^{2} + b^{2} + c^{2}} |

Equation of plane parallel to x + 2y - 2z = d is
x + 2y -2z - d = 0 …………………………(1)
Perpendicular distance from (1,-2,1) is S
$S = | \frac{1 - 4 - 2 + d}{\sqrt{1 + 4 + 4}} | \Rightarrow d = 10$

The Plane exercise fill in the blank question 2

Answer:
10
Hint:
Two perpendicular plane equals 0
Given:
3x - 6y - 2z = 7 and 2x + y - λz = 5
Solution:
The equations of planes are
3x - 6y - 2z -7 =0
and 2x + y - kz - 5 = 0
Since two planes are perpendicular to each other
∴(3)(2) + (-6)(1) + (-2) (-k) = 0 ? a₁a₂+b₁b₂+c₁c₂=0
∴6 - 6 + 2k = 0 2k = 0
k = 0

The Plane exercise fill in the blank question 3

Answer:
2X + 3Y - Y2 + 4 = 0
Hint:
Parallel plane equation also equals to 0
Given:
(1,2,3) and parallel plane 2x + 3y - 4 = 0
Solution:
Equation of parallel to2x + 3y - y2 = 0 is given by
2x + 3y - y2 + λ = 0
Also It passes through (1,2,3)
? 2(1) + 3(2) - 4(3) + λ = 0 ? λ = 4
∴ required plane is 2x + 3y - y2 + 4 = 0

The Plane exercise fill in the blank question 4

Answer:
$\frac{X - 1}{2} = \frac{Y - 2}{3} = \frac{Z - 3}{6}$
Hint:
Use bracket method
Given:
(1,2,3) and normal to the plane 2x - 3y + 6z = 11
Solution:
Let the equation be
$\frac{X - 1}{2} = \frac{Y - 2}{3} = \frac{Z - 3}{6} = 0$
$\sqrt{(1)^{2} + (\frac{3}{2}) + (\frac{6}{3})}$
? we will get normal equation
? 2x - 3y + 6z = 11

The Plane exercise fill in the blank question 5

Answer:
$k = \frac{1}{2}$
Hint:
Use vector 3 dimension
Given:
2x + 3y - 4k = 24 on the coordinate is 8
Solution:

∴

sum of intercepts
=2 + 3 - 4k = 24

The Plane exercise fill in the blank question 6

Answer:
3, -9
Hint:
Solve using fraction method.
Given:
(1,1,1) is half distance fromx + 2y + z + k = 0
Solution:
$\begin{aligned} \sqrt{3} = \frac{1}{2} [\frac{3 + k}{\sqrt{3}}] \\ 3 = \pm \frac{1}{2} (3 + k) \\ 3 + k = \pm 6 \\ 3 + k = 6 3 + k = 6 \\ k = 3 k = - 9 \end{aligned}$

The Plane exercise fill in the blank question 7

Answer:
(a,b,c)
Hint:
Let the points a,b,c and try to get the equation
Given:
$\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 3$
Solution:
Let the point be (a,b,c)
The plane meets the coordinates are at A,B and C such that centroid of the triangle ABC is (a,b,c)
So, the plane cuts x- axis at (3a,0,0) , so, x- intercept = 3a
The plane cuts z- axis at (0,0,3c)?z- intercept =3c
The plane cuts y-axis at (0,3b,0)?y- intercept =3b
? all one getting the equation
$\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 3$
∴ the point will be (a,b,c)

The Plane exercise fill in the blank question 8

Answer:
ax + by + cz = a2 + b2 + c2
Hint:
Firs find the co-ordinates
Given:
p(a,b,c) perpendicular to op
Solution:
The coordinates of the points o and p ,are (0,0,0) and (a,b,c) respectively. Therefore the direction rahos of op are (a-0)=a,(b-0) = b, and (c - 0) = c
It is known that the equation of the plane passing through the point (x,y,z) is
a(x - x₁) + b(y - y₁)+ c(z - z₁) = 0
∴ the required equation will be
a(x - a) + b(y - b) + c(z - c) = 0
ax - a² + by - b² + cz - c² = 0
ax + by + cz = a² + b² + c²

The Plane exercise fill in the blank question 9

Answer:
(2,2,2)
Hint:
Let its intercept be λ
Given:
Sum of the reciprocals of its intercepts on the coordinates axis is 1/2
Solution:
Let the equation of the variable plane be

\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = λ \dots \dots \dots \dots (1)

The intercepts on the coordinate axes are a,b,c
Let the sum of reciprocals of intercept in constant =λ
Therefore,

\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = λ \Rightarrow \frac{(\frac{1}{λ})}{a} + \frac{(\frac{1}{λ})}{b} + \frac{(\frac{1}{λ})}{c} = λ

(\frac{1}{λ}, \frac{1}{λ}, \frac{1}{λ})

lies on the plane (1)
Hence, the variable plane (1) always passes through
the fixed point

(\frac{1}{(\frac{1}{2})}, \frac{1}{(\frac{1}{2})}, \frac{1}{(\frac{1}{2})} \Rightarrow 2, 2, 2)

The Plane exercise fill in the blank question 10

Answer:
1
Hint:
Use formulae of circumantce after finding c0-ordinates
Given:
x + αy +2 = 5
Solution:
Given equation of plane

x + α y + 2 = 5 \Rightarrow \frac{x}{5} + \frac{α y}{5} + \frac{z}{5} = 1

Comparing above equation with equation of plane in intercept from

\frac{x}{x_{1}} + \frac{y}{y_{1}} + \frac{z}{z_{1}} = 1

x_{1} = 5, y_{1} = 5, z_{1} = 5

So coordinates of points A,B,C?A(5,0,0),B(0,5,0),C(0,0,5)
Let circumentre D(x,y,z)

∴ A D = \sqrt{(x - 5)^{2} + y^{2} + z^{2}}, B D = \sqrt{x^{2} + (y - 5)^{2} + z^{2}}, C D = \sqrt{x^{2} + y^{2} + (z - 5)^{2}}

AD=BD=CD ? taking AD=BD

\Rightarrow \sqrt{(x - 5)^{2} + y^{2} + z^{2}} = \sqrt{x^{2} + (y - 5)^{2} + z^{2}}

On squaring both sides
(x - 5)² + y² + z² = x² + (y - 5)² + z² ? 1 = 1
∴ from equation (1) ,x,y can be x=1, y=1
∴ α=1 (also)(by solving)

The Plane exercise fill in the blank question 11

Answer:
-4
Hint:
Use vector dot product
Given:
$\frac{2 x - 1}{2} = \frac{2 - y}{2} = \frac{2 + z}{k}$
is parallel to the plane 2x - y + z = 3
Solution:

\begin{aligned} \frac{x - x_{1}}{a} = \frac{y - y_{1}}{b} = \frac{z - z_{1}}{c} \\ \Rightarrow \frac{x - \frac{1}{2}}{1} = \frac{y - 2}{- 2} = \frac{z + 1}{9} \\ \vec{x} = 2 \hat{i} - \hat{j} + \hat{k}, \vec{b} = \hat{i} - 2 \hat{j} + \hat{k} \\ \vec{x} . \vec{b} = 0 \Rightarrow 2 + 2 + k = 0 \Rightarrow k = - 4 \end{aligned}

The Plane exercise fill in the blank question 12

Answer:
-6
Hint:
Use direction ratios
Given:
$\frac{x - 2}{3} = \frac{y - 1}{- 5} = \frac{z + 2}{2}$
Solution:
The line is
$\frac{x - 2}{3} = \frac{y - 1}{- 5} = \frac{z + 2}{2}$
The direction radius of the line are(3,-5,2)
As the line lies in the plane x + 3y - αz + 7 = 0
We have

3 - 15 - 2 α + 7 = 0 \Rightarrow - 2 α - 5 = 0

α = - \frac{5}{2}

The Plane exercise Fill in the blank question 13

Answer:
-9
Hint:
Use direction ratios
Given:
$\frac{x + 1}{λ} = \frac{y - 1}{1} = \frac{z + 2}{- 4}$
Solution:
Let the given line is
$\frac{x + 1}{λ} = \frac{y - 1}{1} = \frac{z + 2}{- 4}$
Direction ratios are λ,1,-4
And the given plane 2x+ 2y - 8z = -5
Its direction ratios are 2,2,-4
And the given line and plane are perpendicular, then
λ(1) + 1(1) + (-4)(-8) = 0
2λ + 2 + 16=0
λ = -9

The Plane exercise Fill in the blank question 14

Answer:
6x + 4y + 3z = 12 is the required equation of plane
Hint:
Equation of the plane that cut the coordinate axes =1
Given:
(2,0,0) , (0,3,0) and (0,0,4)
Solution:
We know that, equation of the plane that cut the coordinate axes at
(2,0,0) , (0,3,0) and (0,0,4) is

\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1

Hence, the equation of plane passes through the points (2,0,0) , (0,3,0) and (0,0,4)
is

\frac{x}{2} + \frac{y}{3} + \frac{z}{4} = 1

= 6x + 4y + 3z = 12

The Plane exercise fill in the blank question 15

Answer:
x + y + z = 2 is the required Cartesian product
Hint:
Use vector product
Given:
$\vec{r} . (\hat{i} + \hat{j} + \hat{k}) = 2$
Solution:
We have,

\begin{aligned} \vec{r} . (\hat{i} + \hat{j} + \hat{k}) = 2 \\ \Rightarrow (x \hat{i} + y \hat{j} + z \hat{k}) . (\hat{i} + \hat{j} + 2) = 2 \\ \Rightarrow x + y + z = 2 \end{aligned}

The Plane exercise Fill in the blank question 16

Answer:
$- 2, \frac{4}{3}, \frac{- 4}{5}$
are the intercepts made by the plane
Hint:
Convert the given equation into intercept form
Given:
2x - 3y + 5z + 4 = 0
Solution:
First let’s convert the given equation to intercept form i.e

\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1

Where a,b,c are x,y nad z intercepts respectively
Given,
2x - 3y + 5z + 4 = 0
⇒ -2x + 3y - 5z = 4
Dividing by 4 both sides

\frac{x}{- 2} + \frac{y}{\frac{4}{3}} + \frac{z}{\frac{4}{- 5}} = 1

On comparing we have intercepts as
$- 2, \frac{4}{3}, \frac{- 4}{5}$

The Plane exercise Fill in the blank question 17

Answer:
$s i n^{- 1} [\frac{9}{\sqrt{156}}]$
is the angle between the line and the plane
Hint:
Find

θ

using vector dot product
Given:
$\vec{r} = (5 \hat{i} - \hat{j} - 4 \hat{k}) + λ (2 \hat{i} - \hat{j} - \hat{k}) and the plane \vec{r} . (3 \hat{i} - 4 \hat{j} - \hat{k}) + 5 = 0$
Solution:
Line
$\vec{r} = (5 \hat{i} - \hat{j} - 4 \hat{k}) + λ (2 \hat{i} - \hat{j} - \hat{k})$
Is parallel to the vector

\vec{b} = 2 \hat{i} - \hat{j} + \hat{k}

Normal to the plane is

\vec{n} = 3 \hat{i} - 4 \hat{j} - \hat{k}

Let

θ

is the angle between line and plane
Then,

s i n θ = \frac{| \vec{b} . \vec{n} |}{| \vec{b} | | \vec{n} |} \Rightarrow \frac{| (2 \hat{i} - \hat{j} + \hat{k}) . (3 \hat{i} - 4 \hat{j} + \hat{k}) |}{\sqrt{6} . \sqrt{26}}

= \frac{| 6 + 4 - 1 |}{\sqrt{156}} = \frac{9}{\sqrt{156}}

θ = s i n^{- 1} [\frac{9}{\sqrt{156}}]

The Plane exercise Fill in the blank question 18

Answer:
$\vec{r} . (5 \hat{i} - 3 \hat{j} - 2 \hat{k}) = 38$
is the required equation of plane.
Hint:
Use figure to solve the equation
Given:
(5,-3,-2)
Solution:

From the figure normal to the plane is

\vec{n} = \vec{O P} = (5 \hat{i} - 3 \hat{j} - 2 \hat{k})

Plane passing through the point P(5,-3,-2)
Equation of plane is

\begin{aligned} 5 (x - 5) - 3 (y + 3) - 2 (z + 2) = 0 \\ 5 x - 3 y - 2 z = 38 \end{aligned}

In vector form
$\vec{r} . (5 \hat{i} - 3 \hat{j} - 2 \hat{k}) = 38$

The Plane exercise Fill in the blank question 19

Answer:
3 units is the required distance.
Hint:
use vector dot product to find the distance
Given:
$2 x + y - 2 z - 6 = 0 and 4 x + 2 y - 4 z = 0$
Solution:
Let
$\begin{aligned} P_{1} \to 2 x + y - 2 z - 6 = 0 \\ a x + b y + c z + d_{1} = 0 \\ P_{2} \to 4 x + 2 y - 4 z = 0 \\ a x + b y + c z + d_{2} = 0 \end{aligned}$
$\begin{aligned} ∵ P_{1} | | P_{2} \end{aligned}$
$\begin{aligned} Distance between P_{1} and P_{2} = | \frac{d_{2} - d_{1}}{\sqrt{a^{2} + b^{2} + c^{2}}} | \\ = | \frac{0 - 6}{\sqrt{2^{2} + 1^{2} + (- 2)^{2}}} | = | \frac{- 6}{4 + 1 + 4} | \\ = 3 u n i t s \end{aligned}$

The Plane exercise Fill in the blank question 20

Answer:
x - 3z = 10 is the required Cartesian equation
Hint:
use direction cosine formula.
Given:
P(1,0,-3)
Solution:
Assume direction cosine of the normal (a,b,c)
Equation of plane passing through (x,y,z)

\begin{aligned} \Rightarrow a (x - x_{1}) + b (y - y_{1}) + c (z - z_{1}) = 0 \end{aligned}

Point P(1,0,-3)

\begin{aligned} \Rightarrow a (x - 1) + b (y - 0) + c (z - (- 3)) = 0 \end{aligned}

Direction cosine of normal
(a,b,c) = (1 - 0, 0 - 0, -3 - 0)
(a,b,c) = (1,0,-3)
Equation of the plane
[(x - 1)+ 0 (y - 0) - 3 (2- (-3))]=0
x - 3z = 10

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Frequently Asked Questions (FAQs)

1. How are two planes related to each other?

In a three-dimensional space, two planes can be related in three ways.

1. They can run parallel to one another.

2. They can be the same.

3. They can cross each other.

2. What are the properties of planes?

In a three-dimensional space, if there are two different planes, they are either parallel to each other or intersect in a line.
A line can be parallel to a plane, intersect it at a single point, or exist within the plane.
If two different lines are perpendicular to each other on the same plane, they must be parallel to one another.
If two different planes are perpendicular to a common line, they must be parallel to one another.

3. What exactly is a Plane Figure?

A plane figure is any geometric figure with no thickness.

4. Is it enough to study from RD Sharma Class 12 solutions from an exam point of view?

Rd Sharma Class 12th Exercise FBQ are compiled in a step-by-step manner in simple language for students' ease of comprehension. Referring to these solutions while solving can help students improve their conceptual knowledge.

5. What is the best website for studying RD Sharma Solutions for Class 12 Maths?

On the Career360 website, you can find Rd Sharma Class 12th Exercise FBQ solutions, as well as step-by-step answers to all of the questions in the RD Sharma textbook. As a result, in order to understand the important topics, students in Class 12 should learn all of the concepts covered in the syllabus.