Three Dimensional Geometry

Previous Years' CBSE Board Questions

11.1 Introduction

MCQ

1 Distance of the point <math>(p, q, r)</math> from y-axis is

1. q (b) |q|
2. <math>|q| + |r|</math> (d) <math>\sqrt{p^2+r^2}</math> (2023) U

2 The length of the perpendicular drawn from the point (4, -7, 3) on the y-axis is

1. 3 units (b) 4 units
2. 5 units (d) 7 units (2020) U

3 The vector equation of XY-plane is

1. <math>\vec{r} \cdot \hat{k} = 0</math> (b) <math>\vec{r} \cdot \vec{j} = 0</math>
2. <math>\vec{r} \cdot \hat{i} = 0</math> (d) <math>\vec{r} \cdot \vec{n} = 1</math> (2020) Ap

VSA (1 mark)

Write the distance of a point P(a, b, c) from x-axis. (2020C, Delhi 2014C) (EV)

(2020) An

11.2 Direction Cosines and Direction Ratios of a Line

MCQ

(2019) Ev

1. If the direction cosines of a line are <math>(\frac{1}{a}, \frac{1}{a}, \frac{1}{a})</math>, then
2. 0 < a < 1 (b) a>2
3. a > 0 (d) <math>a = \pm \sqrt{3}</math>

1. If a line makes angles of 90°, 135° and 45° with the x, y and z axes respectively, then its direction cosines are
2. (a) <math>0, -\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}</math><br><math>\frac{1}{\sqrt{2}}, 0, -\frac{1}{\sqrt{2}}</math> (b) <math>-\frac{1}{\sqrt{2}}</math>, 0, <math>\frac{1}{\sqrt{2}}</math><br> (d) 0, <math>\frac{1}{\sqrt{2}}</math>, <math>\frac{1}{\sqrt{2}}</math>

V5A (1 mark)

1. Find the direction cosines of a line which makes equal angles with the coordinate axes. (2019) Ev
2. The Cartesian equation of a line AB is :
3. If a line has the direction ratios - 18, 12, - 4, then what are its direction cosines? (2019) Ev
4. If a line makes angles 90°, 135°, 45° with the x, y and z axes respectively, find its direction cosines. (NCERT, Delhi 2019) An

Find the direction cosines of a line parallel to line AB.

(Term II, 2021-22) [EV]

1. If a line makes angles 90° and 60° respectively with the positive directions of x and y axes, find the angle which it makes with the positive direction of z-axis. (Delhi 2017) Ev

OR If a line makes angles 90°, 60° and <math>\theta</math> with x, y and z-axis respectively, where <math>\theta</math> is acute, then find <math>\theta</math>.

11. If a line makes angles α, β, γ with the positive

direction of coordinate axes, then write the value of

<math>\sin^2\alpha + \sin^2\beta + \sin^2\gamma</math>. (Delhi 2015C) [U

SAI (2 marks)

If a line makes an angle α, β, γ with the coordinate

axes, then find the value of <math>\cos 2\alpha + \cos 2\beta + \cos 2\gamma</math>.

(Term II, 2021-22) [U]

Find all the possible vectors of magnitude 5√3 which

are equally inclined to the coordinate axes.

(Term II, 2021-22) U

11.3 Equation of a Line in Space

VSA (1 mark)

The vector equation of a line which passes through

the points (3, 4, -7) and (1, -1, 6) is _____

A line passes through the point with position vector

<math>2\hat{i} - \hat{j} + 4\hat{k}</math> and is in the direction of the vector <math>\hat{i} + \hat{j} - 2\hat{k}</math>.

Find the equation of the line in cartesian form.

The equation of a line are <math>5x - 3 = 15y + 7 = 3 - 10z</math>.

Write the direction cosines of the line. (Al 2015) [EV]

17. If the cartesian equation of a line is <math>\frac{3-x}{c} = \frac{y+4}{7} = \frac{2z-6}{4}</math>,

write the vector equation for the line. (Al 2014) [[v]

SAL (2 marks)

The equations of a line are 5x - 3 = 15y + 7 = 3 - 10z.

Write the direction cosines of the line and find the

coordinates of a point through which it passes. (2023)

Write the cartesian equation of the line PQ passing

through points P(2, 2, 1) and Q(5, 1, -2). Hence, find

the y-coordinate of the point on the line PQ whose

1. coordinate is -2. (Term II, 2021-22) Ev

<math>\frac{2x-1}{12} = \frac{y+2}{2} = \frac{z-3}{3}</math>

The x-coordinate of a point on the line joining

the points P(2, 2, 1) and Q(5, 1, -2) is 4. Find its

1. coordinate. (AI 2017) Ap

SAII (3 marks)

Find the coordinates of the point where the line

through the points (1, 1, 8) and (5, 2, 10) crosses the

ZX-plane. (Term II, 2021-22C) (Ap)

(Delhi 2015) Ev

1. If a line makes 60° and 45° angles with the positive directions of x-axis and z-axis respectively, then find the angle that it makes with the positive direction of
2. axis. Hence, write the direction cosines of the line.

(Term II, 2021-22) [FV]

(4 marks)

Prove that the line through A(0, -1, -1) and B(4, 5, 1) intersects the line through C(3, 9, 4) and D(-4, 4, 4). (Foreign 2016)

1. Show that the lines <math>\frac{x+1}{3} = \frac{y+3}{5} = \frac{z+5}{7}</math> and <math>\frac{x-2}{1} = \frac{y-4}{2} = \frac{z-6}{5}</math> intersect. Also find their point of intersection. (Delhi 2014) Ev
2. Show that lines <math>\vec{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(3\hat{i} - \hat{j})</math> and <math>\vec{r} = (4\hat{i} - \hat{k}) + \mu(2\hat{i} + 3\hat{k})</math> intersect. Also, find their point of intersection. (Delhi 2014) Ev

_A II (5/6 marks)

A line with direction ratios < 2, 2, 1 > intersects the lines <math>\frac{x-7}{3} = \frac{y-5}{2} = \frac{z-3}{1}</math> and <math>\frac{x-1}{2} = \frac{y+1}{4} = \frac{z+1}{3}</math> at the points P and Q respectively. Find the length and the equation of the intercept PQ. (2019C)

11.4 Angle between Two Lines

MCQ

1. no. 28 is Assertion and Reason based question carrying

1 mark. Two statements are given, one labelled Assertion

1. and the other labelled Reason (R). Select the correct

answer from the codes (a), (b), (c) and (d) as given below.

1. Assertion (A): The lines <math>\vec{r} = \vec{a}_1 + \lambda \vec{b}_1</math> and <math>\vec{r} = \vec{a}_2 + \mu \vec{b}_2</math> are perpendicular, when <math>\vec{b}_1 \cdot \vec{b}_2 = 0</math>. Reason (R): The angle <math>\theta</math> between the lines <math>\vec{r} = \vec{a}_1 + \lambda \vec{b}_1</math> and <math>\vec{r} = \vec{a}_2 + \mu \vec{b}_2</math> is given by <math>\cos \theta = \frac{\vec{b}_1 \cdot \vec{b}_2}{|\vec{b}_1| |\vec{b}_2|}</math>.

1. Both A and R are true and R is the correct explanation of A.

1. Both A and R are true and R is not the correct explanation of A.
2. A is true but R is false.
3. A is false but R is true.

(2023)

The angle between the lines 2x = 3y = -z and 6x = -y <math>= -4z</math> is

1. 00 (b) 30° (c) 45° (d) 90°

(2023)

1. If the two lines <math>L_1: x = 5, \frac{y}{3-\alpha} = \frac{z}{-2}</math> <math>L_2: x = 2, \frac{y}{-1} = \frac{z}{2-\alpha}</math> are perpendicular, then the value of <math>\alpha</math> is
2. 2 3 (b) 3 (c) 4 (d) <math>\frac{7}{3}</math>

VSA (1 mark)

Find the vector equation of the line which passes

through the point (3, 4, 5) and is parallel to the vector

2î+2ĵ-3k · (Delhi 2019) (EV)

Find the cartesian equation of the line which passes

through the point (-2, 4, -5) and is parallel to the line

<math>\frac{x+3}{3} = \frac{4-y}{5} = \frac{z+8}{6}</math>. (Delhi 2019) (Ev

Find the angle between the lines

<math>\vec{r} = 2i - 5j + k + \lambda(3i + 2j + 6k)</math> and

<math>\vec{r} = 7\hat{i} - 6\hat{k} + \mu(\hat{i} + 2\hat{j} + 2\hat{k}).</math> (Foreign 2014) Ap

34. Write the equation of the straight line through the

point <math>(\alpha, \beta, \gamma)</math> and parallel to z-axis. (AI 2014C) (AD

SAI (2 marks)

Find the vector equation of the line passing through

the point (2, 1, 3) and perpendicular to both the lines

<math>\frac{x-1}{1} = \frac{y-2}{2} = \frac{z-3}{2}; \frac{x}{2} = \frac{y}{2} = \frac{z}{5}.</math> (2023)

Find the vector and the cartesian equations of a line

that passes through the point A(1, 2, -1) and parallel

to the line <math>5x - 25 = 14 - 7y = 35z</math>. (2023)

Find the value of k so that the lines x = -y = kz and

<math>x-2=2y+1=-z+1</math> are perpendicular to each other.

38. Find the vector equation of the line passing

through the point A(1, 2, -1) and parallel to the line

<math>5x - 25 = 14 - 7y = 35z</math>. (Delhi 2017) (Ap

SA II (3 marks)

1. Find the coordinates of the foot of the perpendicular drawn from point (5, 7, 3) to the line <math>\frac{x-15}{3} = \frac{y-29}{8} = \frac{z-5}{-5}</math>. (2023)

Find the coordinates of the foot of the perpendicular

drawn from the point P(0, 2, 3) to the line

<math>\frac{x+3}{5} = \frac{y-1}{2} = \frac{z+4}{3}</math> (2023)

LAI (4 marks)

Find the value of λ, so that the lines

<math>\frac{1-x}{3} = \frac{7y-14}{\lambda} = \frac{z-3}{2}</math> and <math>\frac{7-7x}{3\lambda} = \frac{y-5}{1} = \frac{6-z}{5}</math>

are at right angles. Also, find whether the lines are

intersecting or not. (Delhi 2019) Ev

Find the vector and cartesian equations of the line

through the point (1, 2, -4) and perpendicular to the

two lines <math>\vec{r} = (8\hat{i} - 19\hat{j} + 10\hat{k}) + \lambda(3\hat{i} - 16\hat{j} + 7\hat{k})</math> and

<math>\vec{r} = (15\hat{i} + 29\hat{j} + 5\hat{k}) + \mu(3\hat{i} + 8\hat{j} - 5\hat{k}).</math>

(Delhi 2016, Al 2015) 🕼

Find the vector and cartesian equations of the line

passing through the point (2, 1, 3) and perpendicular

(2020C) Ap