# NCERT Solutions for Class 12 Maths Maths Part-2 Chapter 11

## Three Dimensional Geometry Class 12

### Exercise 11.1 : Solutions of Questions on Page Number : 467

Q1 :

If a line makes angles 90°, 135°, 45° with x, y and z-axes respectively, find its direction cosines.

Let direction cosines of the line be l, m, and n.

Therefore, the direction cosines of the line are

Q2 :

Find the direction cosines of a line which makes equal angles with the coordinate axes.

Let the direction cosines of the line make an angle α with each of the coordinate axes.

l= cos α, m = cos α, n = cos α

Thus, the direction cosines of the line, which is equally inclined to the coordinate axes, are

Q3 :

If a line has the direction ratios -18, 12, -4, then what are its direction cosines?

Q4 :

Show that the points (2, 3, 4), (-1, -2, 1), (5, 8, 7) are collinear.

Q5 :

Find the direction cosines of the sides of the triangle whose vertices are (3, 5, - 4), (- 1, 1, 2) and (- 5, - 5, - 2)

### Exercise 11.2 : Solutions of Questions on Page Number : 477

Q1 :

Show that the three lines with direction cosines

are mutually perpendicular.

Q2 :

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).

Q3 :

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

Q4 :

Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector.

Q5 :

Find the equation of the line in vector and in Cartesian form that passes through the point with position vector and is in the direction .

Q6 :

Find the Cartesian equation of the line which passes through the point

( - 2, 4, - 5) and parallel to the line given by

Q7 :

The Cartesian equation of a line is . Write its vector form.

Q8 :

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

Q9 :

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

Q10 :

Find the angle between the following pairs of lines:

(i)

(ii) and

Q11 :

Find the angle between the following pairs of lines:

(i)

(ii)

Q12 :

Find the values of p so the line and

are at right angles.

Q13 :

Show that the lines and are perpendicular to each other.

Q14 :

Find the shortest distance between the lines

Q15 :

Find the shortest distance between the lines and

Q16 :

Find the shortest distance between the lines whose vector equations are

Q17 :

Find the shortest distance between the lines whose vector equations are

### Exercise 11.3 : Solutions of Questions on Page Number : 493

Q1 :

In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin.

(a)z = 2 (b)

(c) (d)5y + 8 = 0

Q2 :

Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector.

Q3 :

Find the Cartesian equation of the following planes:

(a) (b)

(c)

Q4 :

In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.

(a) (b)

(c) (d)

Q5 :

Find the vector and Cartesian equation of the planes

(a) that passes through the point (1, 0, - 2) and the normal to the plane is .

(b) that passes through the point (1, 4, 6) and the normal vector to the plane is .

Q6 :

Find the equations of the planes that passes through three points.

(a) (1, 1, -1), (6, 4, -5), (-4, -2, 3)

(b) (1, 1, 0), (1, 2, 1), (-2, 2, -1)

Q7 :

Find the intercepts cut off by the plane

Q8 :

Find the equation of the plane with intercept 3 on the y-axis and parallel to ZOX plane.

Q9 :

Find the equation of the plane through the intersection of the planes and and the point (2, 2, 1)

Q10 :

Find the vector equation of the plane passing through the intersection of the planes and through the point (2, 1, 3)

Q11 :

Find the equation of the plane through the line of intersection of the planes and which is perpendicular to the plane

Q12 :

Find the angle between the planes whose vector equations are

and .

Q13 :

In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.

(a)

(b)

(c)

(d)

(e)

Q14 :

In the following cases, find the distance of each of the given points from the corresponding given plane.

Point Plane

(a) (0, 0, 0)

(b) (3, - 2, 1)

(c) (2, 3, - 5)

(d) ( - 6, 0, 0)

### Exercise Miscellaneous : Solutions of Questions on Page Number : 497

Q1 :

Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3, 5, -1), (4, 3, -1).

Q2 :

If l1, m1, n1 and l2, m2, n2 are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are m1n2 - m2n1, n1l2 - n2l1, l1m2 ­- l2m1.

Q3 :

Find the angle between the lines whose direction ratios are a, b, c and b - c,

c - a, a - b.

Q4 :

Find the equation of a line parallel to x-axis and passing through the origin.

Q5 :

If the coordinates of the points A, B, C, D be (1, 2, 3), (4, 5, 7), (­-4, 3, -6) and (2, 9, 2) respectively, then find the angle between the lines AB and CD.

Q6 :

If the lines and are perpendicular, find the value of k.

Q7 :

Find the vector equation of the plane passing through (1, 2, 3) and perpendicular to the plane

Q8 :

Find the equation of the plane passing through (a, b, c) and parallel to the plane

Q9 :

Find the shortest distance between lines

and.

Q10 :

Find the coordinates of the point where the line through (5, 1, 6) and

(3, 4, 1) crosses the YZ-plane

Q11 :

Find the coordinates of the point where the line through (5, 1, 6) and

(3, 4, 1) crosses the ZX - plane.

Q12 :

Find the coordinates of the point where the line through (3, ­-4, -5) and (2, - 3, 1) crosses the plane 2x + y + z = 7).

Q13 :

Find the equation of the plane passing through the point (-1, 3, 2) and perpendicular to each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0.

Q14 :

If the points (1, 1, p) and ( - 3, 0, 1) be equidistant from the plane , then find the value of p.

Q15 :

Find the equation of the plane passing through the line of intersection of the planes and and parallel to x-axis.

Q16 :

If O be the origin and the coordinates of P be (1, 2, -3), then find the equation of the plane passing through P and perpendicular to OP.

Q17 :

Find the equation of the plane which contains the line of intersection of the planes , and which is perpendicular to the plane .

Q18 :

Find the distance of the point ( - 1, - 5, - 10) from the point of intersection of the line and the plane.

Q19 :

Find the vector equation of the line passing through (1, 2, 3) and parallel to the planes and .

Q20 :

Find the vector equation of the line passing through the point (1, 2, - 4) and perpendicular to the two lines:

Q21 :

Prove that if a plane has the intercepts a, b, c and is at a distance of P units from the origin, then

Q22 :

Distance between the two planes: and is

(A)2 units (B)4 units (C)8 units

(D)

Q23 :

The planes: 2x - y + 4z = 5 and 5x - 2.5y + 10z = 6 are

(A) Perpendicular (B) Parallel (C) intersect y-axis

(C) passes through