NCERT Solutions For Class 11 Maths chapter-12 Introduction to Three Dimensional Geometry Exercise 12.3

NCERT Solutions for Class-11 Maths Chapter-12 Introduction to Three Dimensional Geometry

NCERT Solutions for Class 11 Maths Chapter-11 Conic Sections Exercise 12.3 is prepared by the expert of Physics Wallah score more with Physics Wallah NCERT Class 11 maths solutions. You can download and share NCERT Solutions for Class 11 Maths.

 

NCERT Solutions for Class-11 Maths Exercise 12.3

 

In each of the following Exercises 1 to 4, find the equations :

 

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 externally.

Solution :
(i) The coordinates of point R that divides the line segment joining points P (x₁,y₁,z₁) and Q (x₂,y₂,z₂) internally in the ratiom:n are

.

Let R (x, y,z) be the point that divides the line segment joining points(–2, 3, 5) and (1, –4, 6) internally in the ratio 2:3

Thus, the coordinates of the required point are.

(ii) The coordinates of point R that divides the line segment joining points P (x₁,y₁,z₁) and Q (x₂,y₂,z₂) externally in the ratiom:n are

.

Let R (x, y,z) be the point that divides the line segment joining points(–2, 3, 5) and (1, –4, 6) externally in the ratio 2:3

Thus, the coordinates of the required point are (–8, 17, 3).

Question2. 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 point Q (5, 4, –6) divide the line segment joining points P (3, 2, –4) and R (9, 8, –10) in the ratiok:1.

Therefore, by section formula,

Thus, point Q divides PR in the ratio 1:2.

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

Solution :
Let the YZ planedivide the line segment joining points (–2, 4, 7) and (3, –5, 8) in the ratiok:1.

Hence, by section formula, the coordinates of point of intersection are given by

On the Y Z plane, thex-coordinate of any point is zero.

Thus, the YZ plane divides the line segment formed by joining the given points in the ratio 2:3.

Question4. Using section formula, show that the points A (2, –3, 4), B (–1, 2, 1) and are collinear.

Solution :
The given points are A (2, –3, 4), B (–1, 2, 1), and.

Let P be a point that divides AB in the ratio k:1.

Hence, by section formula, the coordinates of P are given by

Now, we find the value ofk at which point P coincides with point C.

By taking, we obtaink = 2.

Fork = 2, the coordinates of point P are.

i.e., is a point that divides AB externally in the ratio 2:1 and is the same as point P.

Hence, points A, B, and C are collinear.

Question5. 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 A and B be the points that trisect the line segment joining points P (4, 2, –6) and Q (10, –16, 6)

Point A divides PQ in the ratio 1:2. Therefore, by section formula, the coordinates of point A are given by

Point B divides PQ in the ratio 2:1. Therefore, by section formula, the coordinates of point B are given by

Thus, (6, –4, –2) and (8, –10, 2) are the points that trisect the line segment joining points P (4, 2, –6) and Q (10, –16, 6).