In what ratio does the line y 3 divides the line joining of points the line segment joining a 2 6 and B (- 12 1 find the coordinates of the point of intersection?

In what ratio does y axis divide the line segment joining the points 4,7 and 3, 7 ? [CBSE 2012]

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Let the line x-y-2=0  divide the line segment joining the points A (3,1) and B (8,9)  in the ratio k : 1 at P.

Then, the coordinates of P are

`p ((8k+3)/(k+1),(9k-1)/(k+1))`

Since, P lies on the line  x - y -2 =0 we have:

` ((8k+3)/(k+1)) - ((9k-1)/(k+1)) -2=0`

⇒ 8k + 3- 9k + 1- 2k - 2 = 0 

⇒ 8k -9k -2k +3+1 - 2 = 0

⇒ -3k +2 = 0

⇒ - 3k=-2

`⇒ k =2/3`

So, the required ratio is  `2/3:1 `which is equal to 2 : 3.

Solution:

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 m₁: m₂ is given by the Section Formula: P(x, y) = [(mx₂ + nx₁) / m + n, (my₂ + ny₁) / m + n]

Let the ratio in which the line segment joining A(- 3, 10) and B(6, - 8) be divided by point C(- 1, 6) be k : 1.

By Section formula, C(x, y) = [(mx₂ + nx₁) / m + n, (my₂ + ny₁) / m + n]

m = k, n = 1

Therefore,

- 1 = (6k - 3) / (k + 1)

- k - 1 = 6k - 3

7k = 2

k = 2 / 7

Hence, the point C divides line segment AB in the ratio 2 : 7.

☛ Check: NCERT Solutions for Class 10 Maths Chapter 7

Video Solution:

NCERT Class 10 Maths Solutions Chapter 7 Exercise 7.2 Question 4

Summary:

The ratio in which the line segment joining the points (- 3, 10) and (6, - 8) is divided by (- 1, 6) is 2 : 7.

☛ Related Questions:

• Find the ratio in which the line segment joining A (1, - 5) and B (- 4, 5) is divided by the x-axis. Also, find the coordinates of the point of division.
• If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find x and y.
• Find the coordinates of a point A, where AB is the diameter of a circle whose centre is (2, - 3) and B is (1, 4).
• If A and B are (- 2, - 2) and (2, - 4), respectively, find the coordinates of P such that AP = 3/7 AB and P lies on the line segment AB.

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