## NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry Ex 11.2

Get Free NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry Ex 11.2 PDF in Hindi and English Medium. Sets Class 12 Maths NCERT Solutions are extremely helpful while doing your homework. Three Dimensional Geometry Exercise 11.2 Class 12 Maths NCERT Solutions were prepared by Experienced LearnCBSE.in Teachers. Detailed answers of all the questions in Chapter 11 Class 12 Three Dimensional Geometry Ex 11.2 provided in NCERT Textbook.

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### NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry Ex 11.2

**Ex 11.2 Class 12 Maths Question 1.**

Show that the three lines with direction cosines:

(frac { 12 }{ 13 } ,frac { -3 }{ 13 } ,frac { -4 }{ 13 } ,frac { 4 }{ 13 } ,frac { 12 }{ 13 } ,frac { 3 }{ 13 } ,frac { 3 }{ 13 } ,frac { -4 }{ 13 } ,frac { 12 }{ 13 } )

are mutually perpendicular.

**Solution:**

Let the lines be L1,L2 and L3.

∴ For lines L1 and L2

**Ex 11.2 Class 12 Maths Question 2.**

Show that the line through the points (1,-1,2) (3,4, -2) is perpendicular to the line through the points (0,3,2) and (3,5,6).

**Solution:**

Let A, B be the points (1, -1, 2), (3, 4, -2) respectively Direction ratios of AB are 2,5, -4

Let C, D be the points (0, 3, 2) and (3, 5, 6) respectively Direction ratios of CD are 3, 2,4 AB is Perpendicular to CD if

**Ex 11.2 Class 12 Maths Question 3.**

Show that the line through the points (4,7,8) (2,3,4) is parallel to the line through the points (-1,-2,1) and (1,2,5).

**Solution:**

Let the points be A(4,7,8), B (2,3,4), C (-1,-2,1) andD(1,2,5).

Now direction ratios of AB are

**Ex 11.2 Class 12 Maths Question 4.**

Find the equation of the line which passes through the point (1,2,3) and is parallel to the vector (3hat { i } +2hat { j } -2hat { k } )

**Solution:**

Equation of the line passing through the point

**Ex 11.2 Class 12 Maths Question 5.**

Find the equation of the line in vector and in cartesian form that passes through the point with position vector (2hat { i } -hat { j } +4hat { k } ) and is in the direction (hat { i } +2hat { j } -hat { k } ).

**Solution:**

The vector equation of a line passing through a point with position vector (overrightarrow { a } ) and parallel to the

**Ex 11.2 Class 12 Maths Question 6.**

Find the cartesian equation of the line which passes through the point (-2,4, -5) and parallel to the line is given by (frac { x+3 }{ 3 } =frac { y-4 }{ 5 } =frac { z+8 }{ 6 } )

**Solution:**

The cartesian equation of the line passing through the point (-2,4, -5) and parallel to the

**Ex 11.2 Class 12 Maths Question 7.**

The cartesian equation of a line is

(frac { x-5 }{ 3 } =frac { y+4 }{ 7 } =frac { z-6 }{ 2 } )

write its vector form.

**Solution:**

The cartesian equation of the line is

(frac { x-5 }{ 3 } =frac { y+4 }{ 7 } =frac { z-6 }{ 2 } )

Clearly (i) passes through the point (5, – 4, 6) and has 3,7,2 as its direction ratios.

=> Line (i) passes through the point A with

**Ex 11.2 Class 12 Maths Question 8.**

Find the vector and the cartesian equations of the lines that passes through the origin and (5,-2,3).

**Solution:**

The line passes through point

(therefore overrightarrow { a } =overrightarrow { 0 } )

Direction ratios of the line passing through the

**Ex 11.2 Class 12 Maths Question 9.**

Find the vector and cartesian equations of the line that passes through the points (3, -2, -5), (3,-2,6).

**Solution:**

The PQ passes through the point P(3, -2, -5)

**Ex 11.2 Class 12 Maths Question 10.**

Find the angle between the following pair of lines

(i) (overrightarrow { r } =2hat { i } -5hat { j } +hat { k } +lambda (3hat { i } +2hat { j } +6hat { k } ))

(andquad overrightarrow { r } =7hat { i } -6hat { j } +mu (hat { i } +2hat { j } +2hat { k } ))

(ii) (overrightarrow { r } =3hat { i } +hat { j } -2hat { k } +lambda (hat { i } -hat { j } -2hat { k } ))

(overrightarrow { r } =2hat { i } -hat { j } -56hat { k } +mu (3hat { i } -5hat { j } -4hat { k } ))

**Solution:**

(i) Let θ be the angle between the given lines.

The given lines are parallel to the vectors

**Ex 11.2 Class 12 Maths Question 11.**

Find the angle between the following pair of lines

(i) (frac { x-2 }{ 2 } =frac { y-1 }{ 5 } =frac { z+3 }{ -3 } andfrac { x+2 }{ -1 } =frac { y-4 }{ 8 } =frac { z-5 }{ 4 } )

(ii) (frac { x }{ 2 } =frac { y }{ 2 } =frac { z }{ 1 } andfrac { x-5 }{ 4 } =frac { y-2 }{ 1 } =frac { z-3 }{ 8 } )

**Solution:**

Given

(i) (frac { x-2 }{ 2 } =frac { y-1 }{ 5 } =frac { z+3 }{ -3 } andfrac { x+2 }{ -1 } =frac { y-4 }{ 8 } =frac { z-5 }{ 4 } )

(ii) (frac { x }{ 2 } =frac { y }{ 2 } =frac { z }{ 1 } andfrac { x-5 }{ 4 } =frac { y-2 }{ 1 } =frac { z-3 }{ 8 } )

**Ex 11.2 Class 12 Maths Question 12.**

Find the values of p so that the lines

(frac { 1-x }{ 3 } =frac { 7y-14 }{ 2p } =frac { z-3 }{ 2 } andfrac { 7-7x }{ 3p } =frac { y-5 }{ 1 } =frac { 6-z }{ 5 } )

are at right angles

**Solution:**

The given equation are not in the standard form

The equation of given lines is

**Ex 11.2 Class 12 Maths Question 13.**

Show that the lines (frac { x-5 }{ 7 } =frac { y+2 }{ -5 } =frac { z }{ 1 } andfrac { x }{ 1 } =frac { y }{ 2 } =frac { z }{ 3 } ) are perpendicular to each other

**Solution:**

Given lines

(frac { x-5 }{ 7 } =frac { y+2 }{ -5 } =frac { z }{ 1 } ) …(i)

(frac { x }{ 1 } =frac { y }{ 2 } =frac { z }{ 3 } ) …(ii)

**Ex 11.2 Class 12 Maths Question 14.**

Find the shortest distance between the lines

(overrightarrow { r } =(hat { i } +2hat { j } +hat { k } )+lambda (hat { i } -hat { j } +hat { k } )) and

(overrightarrow { r } =(2hat { i } -hat { j } -hat { k } )+mu (2hat { i } +hat { j } +2hat { k } ))

**Solution:**

The shortest distance between the lines

**Ex 11.2 Class 12 Maths Question 15.**

Find the shortest distance between the lines

(frac { x+1 }{ 7 } =frac { y+1 }{ -6 } =frac { z+1 }{ 1 } andfrac { x-3 }{ 1 } =frac { y-5 }{ -2 } =frac { z-7 }{ 1 } )

**Solution:**

Shortest distance between the lines

**Ex 11.2 Class 12 Maths Question 16.**

Find the distance between die lines whose vector equations are:

(overrightarrow { r } =(hat { i } +2hat { j } +3hat { k) } +lambda (hat { i } -3hat { j } +2hat { k } )) and

(overrightarrow { r } =(4hat { i } +5hat { j } +6hat { k) } +mu (2hat { i } +3hat { j } +hat { k } ))

**Solution:**

Comparing the given equations with

**Ex 11.2 Class 12 Maths Question 17.**

Find the shortest distance between the lines whose vector equations are

(overrightarrow { r } =(1-t)hat { i } +(t-2)hat { j } +(3-2t)hat { k } ) and

(overrightarrow { r } =(s+1)hat { i } +(2s-1)hat { j } -(2s+1)hat { k } )

**Solution:**

Comparing these equation with

### NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry Hindi Medium Ex 11.2

### Class 12 Maths NCERT Solutions

**Chapter 1 Relations and Functions****Chapter 2 Inverse Trigonometric Functions****Chapter 3 Matrices****Chapter 4 Determinants****Chapter 5 Continuity and Differentiability****Chapter 6 Application of Derivatives****Chapter 7 Integrals Ex 7.1****Chapter 8 Application of Integrals****Chapter 9 Differential Equations****Chapter 10 Vector Algebra****Chapter 11 Three Dimensional Geometry****Chapter 12 Linear Programming****Chapter 13 Probability Ex 13.1**

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