## Kerala Plus One Maths Chapter Wise Questions and Answers Chapter 12 Introduction to Three Dimensional Geometry

**Short Answer** **Type Questions **

**(Score 3)**

Question 1.

Distance of the point (x, y, z) from the XY plane is

a. x

b. y

c. z

d. |z|

b. Find the distance of the point (3,4,5) from X-axis.

Answer:

a. d. |z|

x y z are mutually perpendicular line so, the distance of the point (x, y, z) from the x y plane is |z|.

b. On x- axis, coordinate of point will be in the form (3,0,0) so the distance of the point (3, 4, 5) from the x- axis is

=

Question 2.

Consider the points A (3, 1, -2) and B (1,-7,4).

i. In which octants does these points lie.

ii. Find the reflection (image) of the point A in the XY plane.

Answer:

i. A lies in 5^{th} octant (XOYZ’)

B lies in 4^{th} octant (XOY’Z)

ii. (3,1.2)

Question 3.** **a. A point is on the X – axis. What are its y and z-coordinates?

b. L is the foot of the perpendicular drawn from a point P(3,4,5) on the XY-plane. Find the coordinates of point L.

Answer:

a. Coordinates of any point on the X-axis is (x,0,0). Because at X-axis, both y and z corresponding are zero. so ,its y and z-coordinates are zero.

b. Since, in a XY-plane, the z-coordinate will be zero. Hence, the coordinate of the foot of the point L(3,4,0)

Question 4.

Find the distance of point P(3,4, S) from the YZ-plane.

Answer:

When we draw a perpendicular line from the point P(3,4,5) on the YZ-plane, the x -coordinate will be zero and the other coordinates y and z will be 4 and 5, i.e., coordinate on YZ pale be Q (0,4,5).

Distance between P and Q

Question 5.

a.The distance between the X-axis and the point (3,12,5) is

a. 3

b. 13

c. 12

d. 4

b. Consider the point A (2, -1,2), which In octant

Answer:

a. 13

Let point on the x – axis be (a, 0, 0) So the distance between (3, 0,0) and (3,12,5) is

b. fourth

Question 6.

a If a parallelopiped formed by planes drawn through the points (5,8,10) and (3,6,8) parallel to the corrdinates planes, their find the length of diagonal of the parallelopiped.

a.

b.

c.

d.

b. Find the coordinates of the point which divides the join of the points A (2, -1,3) and B (4,3,1) externally in the ratio 3:4.

Answer:

Question 7.

a. Verify whether the points (-1,2,1) (1,-2, 5), (4, -7,8) and (2, -3,4) are the vertices of a parallelogram.

b. The coordinates of the point R which divided the line segment joining P(x_{1}, y_{1}, z_{1}) and Q(x_{2}, y_{2}, z_{2}) internally in the ratio m: n are given by……………..

Answer:

a. Let the points be A, B, C and D respectively.

Using distance formula,

AB = 6,BC = ,CD=6 and AD=

AB = CD and BC=AD

∴ ABCD is a parallelogram.

Question 8.

Consider the points A (-2, 4, 7) and B (3,-5,8).

i. If P divides AB in the ratio k: 1, then find the coordinates of P.

ii. Find the co-ordinates of the point where the line segment AB crosses the YZ – plane.

Answer:

Question 9.

Consider the points A (-2,3, 5), B (1, 2,3) and C (7,0,-1).

i. Using the distance formula, show that the points A, B and C are collinear.

ii. Find the ratio in which B divides the line segment AC.

Answer:

Question 10.

Find the ratio in which the line joining (2,4, 5), (3,5, -4) is divided by the YZ plane.

Answer:

Let the yz – plane divide the line segment joining the points (2,4,5) and (3,5, -4) at the point P in the ratio k: 1. Then the coordinate of P is

**Short Answer** **Type Questions **

**(Score 4)**

Question 1.

a Find the locus of a point which moves so that its distances from the points

(3,4, -5) and (-2,1,4) are equal.

b. Show that the points A (10,7,0) B(6,6, -1) C (6, 9, – 4) form a right angled isosceles triangle.

Answer:

a. Let P (x, y, z) be a point on the locus. PA=PB PA^{2} = PB^{2 }(x-3)^{2} + (y-4)^{2} + (z + 5)^{2 }= (x + 2)^{2} + (y- 1)^{2} + (z-4)^{2 }= 10x + 6y- 18z-29 = 0

b. Using distance formula,

AB= ,BC= ,AC=

AB = BC (triangle is isosceles)

AC^{2} = AB^{2} + BC^{2} (triangle is right angled) So it is a tight angled isosceles triangle.

Question 2.

Plane XOZ divides the join of (1, -1, 5) and (2,3,4) in the ratio X : 1, then X is

a. -3

b. 1/3

c. 3

d. -1/3

b. Find the ratio in which the plane y -1 = 0 divides the straight line joining (1, -1,3) and (-2,5,4).

Answer:

Question 3.

If P (2,1,5) and Q (0, -5, -1) are two points in space. Find the coordinates of the points M, R, S where

i. M is the midpoint of PQ.

ii. R is the point divides PQ internally in the ratio 2:1

iii. S is the point dividing PQ externally in the ratio 2 :1

Answer:

Question 4.

Consider A ABC with vertices A (0,0,6), B (0,4,0) and C (6,0,0).

i. Find the centroid of Δ ABC.

ii. Find the lengths of the medians.

Answer:

Question 5.

Consider the triangle with vertices A (0,7,-10), B (1,6,-6), C (4,9,-6).

i. Find the sides AB,BC, AC

ii. Prove that the triangle is right angled.

iii. Find the centroid of the triangle

Answer:

Question 6.

Consider the point A (2, -1,2)

i. Find the distance between origin and A.

ii. B (3,4,5) is a point in first octant, find the ratio in which YZ plane divides AB.

iii. Also find the point of division.

Answer:

Question 7.

Points (4,7,8), (2,3,4), (-1, -2,1) and (1,2, 5) are the vertices of a

a. Parallelogram

b. Square

c. Rectangle

d. Rhombus

Answer:

CD of BC = AD

So opposite sides are equal so the given points are vertices of a parallelogram.

**Long Answer** **Type Questions **

**(Score 6)**

Question 1.

a. Consider the points A(3, -2,4), B (1,1,1) and C (-1,4, -2). Using distance formula prove the points are collinear. h Find the coordinates of the points which trisect the line segment joining the points P (4,2, -6) and Q (10, -16,6).

Answer:

Question 2.

Consider the points A (3,2, – 4), B (5,4, -6) and C (9,8,-10).

i. BC and AC and show that A,B,C are collinear.

ii. Find the ratio in which B divides AC using distance formula.

iii. Verify the result using section formula.

Answer:

Question 3.

a. Three vertices of a parallelogram ABCD are A(3, -1,2), B (1,2, – 4) and C (-1,1,2). Find the coordinates of the fourth vertex, h Find the ratio in which the join of the points P (2, -1,3) and Q (4,3,1) is divided

by the points

Answer:

Question 4.

a. Find the co- ordinates of the points which trisect the line segment joining the points P (4,0,1) and Q (2,4,0).

b. Find the locus of the set of points p such that the distance from A (2,3,4) is equal to twice the distance from B (-2,1,2).

Answer:

a. Let the ratio be 1:2 or 2:1

Question 5.

Find the length of the medians of the triangle with vertices A(0,0,6), B(0,4,0) and C(6,0,0).

Answer:

ABC is a triangle with vertices A(0,0,6), B(0,4,0) and C (6,0,0).

Let points D, E and F are the mid-points of BC, AC and AB, respectively. So, AD, BE and CF will be the medians of the triangle.

Question 6.

a. What are the coordinates of the vertices of a cube whose edge is 2 units, one of whose vertices coincides with the origin and three edges passing through the origin coincides with the positive direction of the axes through the origin.

b. Show that the points (-2,6, -2) ,(0,4, -1), (-2,3,1) and (-4,5,0) are the vertices of a square.

Answer:

a. Given, edge of a cube is 2 unit. It is clear that

coordinates of O(0,0,0)

coordinates of A(2,0,0)

coordinates of B(0,2,0)

coordinates of D(0,0,2)

coordinates of B(2,2,0)

coordinates of F(2,2,2)

coordinates of E(2,0,2)

coordinates of C(0,2,2)

**NCERT Questions and Answers**

Question 1.

Find the octant in which the points (-3,1,2) and (-3,1,-2) lie.

Answer:

The point (-3,1,2) lies in second octant and the point (-3,1,-2) lies in octant VI.

Question 2.

**Fill in the blanks:**

i. The x-axis and y-axis taken together determine a plane known as…….

ii. The coordinates of points in the XY-plane are of the form………….

iii. Coordinate planes divide the space into …………….

Answer:

i. XY – plane

ii. (x,y, 0)

iii. Eight

Question 3.

Name the octants in which the following points lie:

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

(- 4,2, -5), (- 4,2,5), (-3, -1,6) (2, – 4, -7).

Answer:

I, IV, VIII, V, VI, II, III, W

Question 4.

Find the distance between the points P (1, -3,4) and Q(-4,1,2).

Answer:

The distance PQ between the points P (1,-3,4) and Q(-4,1,2)

Question 5.

Consider the points A (-2, 3, 5), B (1, 2,3) and C (7,0,-1).

i. Find the ratio in which C divides AB.

ii. Say whether the division is internal or external.

Answer:

Suppose C divides AB in the ratio m: 1

Question 6.

Are the points A (3,6,9), B (10,20,30) and C (25, – 41,5), the vertices of a right angled triangle?

Answer:

By the distance formula, we have

AB^{2}= (10 – 3)^{2} + (20 – 6)^{2} + (30 – 9)^{2 }= 49+196 + 441=686

BC^{2} =(25-10)^{2} +(-41-20)^{2} + (5-30)^{2 }= 225 + 3721+625 = 4571

CA^{2} = (3 – 25)^{2} + (6 + 41)^{2} + (9 – 5)^{2 }= 484 + 2209+16 = 2709

We find that CA^{2} + AB^{2} ≠BC^{2 }Hence, the triangle ABC is not a right angled triangle.

Question 7.

Find the equation of set of points P such that PA^{2}+PB^{2}=2k^{2}, where A and B are the points (3,4,5)

and (-1,3, -7), respectively.

Answer:

Let the coordinates of point P be (x, y, z).

Here PA^{2} = (x – 3)^{2} + (y- 4)^{2} + (z – 5)^{2 }PB^{2} = (x + l)^{2} + (y- 3)^{2} + (z + 7)^{2 }By the given condition PA^{2} + PB^{2} = 2k^{2 }we have (x – 3)^{2} + (y- 4)^{2} + (z – 5)^{2}+ (x + l)^{2 }+ (y-3)^{2} + (z + 7)^{2} = 2k^{2 }i.e., 2x^{2} + 2y^{2} + 2z^{2} – 4x – 14y + 4z = 2k^{2} -109.

Question 8.

Using section formula ,show that the points A (2, -3,4) B (-1,2,1) and C (0, j-, 2) are collinear.

Answer:

Let B divides AC in the ratio k: 1 Coordinates of B is

Question 9.

If the origin is the centroid of ΔPQR with vertices P (2a, 2,6), Q (- 4,3b, -10) and R (8, 14,2c) then find the values of a, b and c.

Answer: