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Textbook | NCERT |

Board | CBSE |

Category | NCERT Solutions |

Class | Class 12 |

Subject | Maths |

Chapter | Chapter 11 |

Exercise | Class 12 Maths Chapter 11 Three Dimensional Geometry Exercise 11.2 |

Number of Questions Solved | 17 |

Contents

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

**NCERT TEXTBOOK EXERCISES**

**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 L_{1},L_{2} and L_{3}.

∴ 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 $3\hat { i } +2\hat { j } -2\hat { k }$

**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 $2\hat { i } -\hat { j } +4\hat { k }$ and is in the direction $\hat { i } +2\hat { j } -\hat { k }$.

**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

Undefined control sequence \therefore

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

$and\quad \overrightarrow { r } =7\hat { i } -6\hat { j } +\mu (\hat { i } +2\hat { j } +2\hat { k } )$

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

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

**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) $\frac { x-2 }{ 2 } =\frac { y-1 }{ 5 } =\frac { z+3 }{ -3 } and\frac { x+2 }{ -1 } =\frac { y-4 }{ 8 } =\frac { z-5 }{ 4 }$

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

**Solution:**

Given

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

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

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

Find the values of p so that the lines

$\frac { 1-x }{ 3 } =\frac { 7y-14 }{ 2p } =\frac { z-3 }{ 2 } and\frac { 7-7x }{ 3p } =\frac { y-5 }{ 1 } =\frac { 6-z }{ 5 }$ 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 $\frac { x-5 }{ 7 } =\frac { y+2 }{ -5 } =\frac { z }{ 1 } and\frac { x }{ 1 } =\frac { y }{ 2 } =\frac { z }{ 3 }$ 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

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

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

**Solution:**

The shortest distance between the lines

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

Find the shortest distance between the lines

$\frac { x+1 }{ 7 } =\frac { y+1 }{ -6 } =\frac { z+1 }{ 1 } and\frac { x-3 }{ 1 } =\frac { y-5 }{ -2 } =\frac { z-7 }{ 1 }$

**Solution:**

Shortest distance between the lines

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

Find the distance between die lines whose vector equations are:

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

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

**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 Exercise 11.2 PDF

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