Chapter 12 Introduction to Three Dimensional Geometry Ex – 12.3

Question 1.

Find the coordinates of the point which divides the line segment joining the points (-2, 3, 5) and (1, -4, 6) in the ratio

(i) 2 : 3 internally,

(ii) 2 : 3 internally

Solution:

(i) Let P(x, y, z) be any point that divides the line segment joining the points A(-2, 3, 5) and B(1, -4, 6) in the ratio 2 : 3 internally.

NCERT Solutions for Class 11 Maths Chapter 12 Introduction to Three Dimensional Geometry Ex 12.3 1
NCERT Solutions for Class 11 Maths Chapter 12 Introduction to Three Dimensional Geometry Ex 12.3 2

(ii) Let P(x, y, z) be any point that divides the line segment joining the points 71 (-2, 3, 5) and B(1, -4, 6) in the ratio 2 : 3 externally. Then

NCERT Solutions for Class 11 Maths Chapter 12 Introduction to Three Dimensional Geometry Ex 12.3 3

Question 2.

Given that P(3, 2, -4), Q(5, 4, -6) and R(9, 8, -10) are collinear. Find the ratio in which Q divides PR.

Solution:

Let Q(5, 4, -6) divide the line segment joining the points P(3, 2, -4) and R(9, 8, -10) in the ratio k : 1 internally.

NCERT Solutions for Class 11 Maths Chapter 12 Introduction to Three Dimensional Geometry Ex 12.3 4

Question 3.

Find the ratio in which the YZ-plane divides the line segment formed by joining the points (-2, 4, 7) and (3, -5, 8).

Solution:

Let the line segment joining the points A(-2, 4, 7) and B(3, -5, 8) be divided by the YZ -plane at a point C in the ratio k : 1.

NCERT Solutions for Class 11 Maths Chapter 12 Introduction to Three Dimensional Geometry Ex 12.3 5
NCERT Solutions for Class 11 Maths Chapter 12 Introduction to Three Dimensional Geometry Ex 12.3 6

Question 4.

Using section formula, show that the points A(2, -3, 4), B(-1, 2, 1), and C\left( 0,\frac { 1 }{ 3 } ,2 \right)  are collinear.

Solution:

Let the points A(2, -3, 4), B(-l, 2,1) and C\left( 0,\frac { 1 }{ 3 } ,2 \right)  be the given points. Let the point P divides AB in the ratio k : 1. Then coordinates of P are \left( \frac { -k+2 }{ k+1 } ,\frac { 2k-3 }{ k+1 } ,\frac { k+4 }{ k+1 } \right)

Let us examine whether, for some value of k, the point P coincides with point C.

NCERT Solutions for Class 11 Maths Chapter 12 Introduction to Three Dimensional Geometry Ex 12.3 7

AB internally in the ratio 2:1. Hence A, B, C are collinear.

Question 5.

Find the coordinates of the points which trisect the line segment joining the points P(4, 2, -6) and Q(10, -16, 6).

Solution:

Let R and S be two points that trisect the line segment PQ.

NCERT Solutions for Class 11 Maths Chapter 12 Introduction to Three Dimensional Geometry Ex 12.3 8