NCERT Solutions Class 11th Maths Chapter 12- Introduction to Three Dimensional Geometry Exercise 12.3

NCERT Solutions Class 11th Maths Chapter 12- Introduction to Three Dimensional Geometry Exercise 12.3

NCERT Solutions Class 11th Maths Chapter 12- Introduction to Three Dimensional Geometry Exercise 12.3

?Chapter – 12?

✍Introduction to Three Dimensional Geometry✍

?Exercise 12.3?

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: Let the line segment joining the points P (-2, 3, 5) and Q (1, -4, 6) be PQ.

(i) 2: 3 internally
By using section formula,

We know that the coordinates of the point R which divides the line segment joining two points P (x₁, y₁, z₁) and Q (x₂, y₂, z₂) internally in the ratio m: n is given by:

Upon comparing we have
x₁ = -2, y₁ = 3, z₁ = 5;
x₂ = 1, y₂ = -4, z₂= 6 and
m = 2, n = 3
So, the coordinates of the point which divides the line segment joining the points P (– 2, 3, 5) and Q (1, – 4, 6) in the ratio 2 : 3 internally is given by:

Hence, the coordinates of the point which divides the line segment joining the points (-2, 3, 5) and (1, -4, 6) is (-4/5, 1/5, 27/5)

(ii) 2: 3 externally
By using section formula,

We know that the coordinates of the point R which divides the line segment joining two points P (x₁, y₁, z₁) and Q (x₂, y₂, z₂) externally in the ratio m: n is given by:

Upon comparing we have
x₁ = -2, y₁ = 3, z₁ = 5;
x₂ = 1, y₂ = -4, z₂ = 6 and
m = 2, n = 3
So, the coordinates of the point which divides the line segment joining the points P (– 2, 3, 5) and Q (1, – 4, 6) in the ratio 2: 3 externally is given by:

∴ The co-ordinates of the point which divides the line segment joining the points (-2, 3, 5) and (1, -4, 6) is (-8, 17, 3).

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 us consider Q divides PR in the ratio k: 1.
By using section formula,

We know that the coordinates of the point R which divides the line segment joining two points P (x₁, y₁, z₁) and Q (x₂, y₂, z₂) internally in the ratio m : n is given by:

Upon comparing we have,
x₁ = 3, y₁ = 2, z₁ = -4;
x₂ = 9, y₂ = 8, z₂ = -10 and
m = k, n = 1
So, we have

9k + 3 = 5 (k+1)
9k + 3 = 5k + 5
9k – 5k = 5 – 3
4k = 2
k = 2/4
= ½
Hence, the ratio in which Q divides PR is 1: 2.

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 formed by joining the points P (-2, 4, 7) and Q (3, -5, 8) be PQ.
We know that any point on the YZ-plane is of the form (0, y, z).
So now, let R (0, y, z) divides the line segment PQ in the ratio k: 1.
Then,
Upon comparing we have,
x₁ = -2, y₁ = 4, z₁ = 7;
x₂ = 3, y₂ = -5, z₂ = 8 and
m = k, n = 1
By using the section formula,

We know that the coordinates of the point R which divides the line segment joining two points P (x₁, y₁, z₁) and Q (x₂, y₂, z₂) internally in the ratio m: n is given by:

So we have,

3k – 2 = 0
3k = 2
k = 2/3
Hence, the ratio in which the YZ-plane divides the line segment formed by joining the points (–2, 4, 7) and (3, –5, 8) is 2:3.

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

Solution: Let the point P divides AB in the ratio k: 1.
Upon comparing we have,
x₁ = 2, y₁ = -3, z₁ = 4;
x₂ = -1, y₂ = 2, z₂ = 1 and
m = k, n = 1
By using section formula,

We know that the coordinates of the point R which divides the line segment joining two points P (x₁, y₁, z₁) and Q (x₂, y₂, z₂) internally in the ratio m : n is given by:

So we have,

Now, we check if for some value of k, the point coincides with the point C.
Put (-k+2)/(k+1) = 0
-k + 2 = 0
k = 2
When k = 2, then (2k-3)/(k+1) = (2(2)-3)/(2+1)
= (4-3)/3
= 1/3
And, (k+4)/(k+1) = (2+4)/(2+1)
= 6/3
= 2
∴ C (0, 1/3, 2) is a point which divides AB in the ratio 2: 1 and is same as P.
Hence, A, B, C are collinear.

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 A (x₁, y₁, z₁) and B (x₂, y₂, z₂) trisect the line segment joining the points P (4, 2, -6) and Q (10, -16, 6).

A divides the line segment PQ in the ratio 1: 2.
Upon comparing we have,
x₁ = 4, y₁ = 2, z₁ = -6;
x₂ = 10, y₂ = -16, z₂ = 6 and
m = 1, n = 2
By using the section formula,

We know that the coordinates of the point R which divides the line segment joining two points P (x₁, y₁, z₁) and Q (x₂, y₂, z₂) internally in the ratio m : n is given by:

So we have,

Similarly, we know that B divides the line segment PQ in the ratio 2: 1.
Upon comparing we have,
x₁ = 4, y₁ = 2, z₁ = -6;
x₂ = 10, y₂ = -16, z₂ = 6 and
m = 2, n = 1
By using the section formula,

We know that the coordinates of the point R which divides the line segment joining two points P (x₁, y₁, z₁) and Q (x₂, y₂, z₂) internally in the ratio m: n is given by:

So we have,

∴ The coordinates of the points which trisect the line segment joining the points P (4, 2, – 6) and Q (10, –16, 6) are (6, -4, -2) and (8, -10, 2).