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Find the ratio in which the line 2x + 3y – 5 = 0 divides the line segment joining the points (8, –9) and (2, 1). Also find the coordinates of the point of division

Solution:

Given, the line 2x + 3y - 5 = 0 divides the line segment joining the points (8, -9) and (2, 1)

We have to find the ratio in which the given line divides the line segment and the coordinates of the point of division.

By section formula,

The coordinates of the point P(x, y) which divides the line segment joining the points A (x₁ , y₁) and B (x₂ , y₂) internally in the ratio k : 1 are [(kx₂ + x₁)/k +1, (ky₂ + y₁)/k + 1]

Here, (x₁ , y₁) = (8, -9) and (x₂ , y₂) = (2, 1)

Let the coordinates of the point be (x, y)

[(k(2) + 8)/k + 1, (k(1) + (-9))/k + 1] = (x, y)

[(2k + 8)/k + 1, (k - 9)/k + 1] = (x, y) ----------------------- (1)

The point lies on the line 2x + 3y - 5 = 0.

Now, 2((2k + 8)/k + 1) + 3((k - 9)/k + 1) = 5

2(2k + 8) + 3(k - 9) = 5(k + 1)

4k + 16 + 3k - 27 = 5k + 5

By grouping,

4k + 3k - 5k = 5 + 27 - 16

7k - 5k = 5 + 11

2k = 16

k = 16/2

k = 8

Therefore, the point divides the line segment in the ratio 8 : 1.

To find the coordinates of the point of division,

Put k = 8 in (1)

(x, y) = [(2(8) + 8)/8 + 1, (8 - 9)/8 + 1]

(x, y) = [(16 + 8)/9, -1/9]

(x, y) = [24/9, -1/9]

(x, y) = [8/3, -1/9]

Therefore, the coordinates of the point of division are 8/3 and -1/9.

✦ Try This: Find the ratio in which the y-axis divides the line segment joining the points (5, -6) and (-1, -4). Also, find the coordinates of the point of division.

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 7

NCERT Exemplar Class 10 Maths Exercise 7.3 Problem 20

Find the ratio in which the line 2x + 3y – 5 = 0 divides the line segment joining the points (8, –9) and (2, 1). Also find the coordinates of the point of division

Summary:

The ratio in which the line 2x + 3y – 5 = 0 divides the line segment joining the points (8, –9) and (2, 1) os 8:1. Also the coordinates of the point of division are 8/3 and -1/9

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