In what ratio does the line divide the line segment joining the points and?

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In what ratio does the origin divide the line segment joining points 5, 0 and 10, 0?

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Solution:

Given, the line segment joining the points (-4, -6) and (-1, 7)

We have to find the ratio of division of 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) = (-4, -6) and (x₂ , y₂) = (-1, 7)

So, [(k(-1) + (-4))/(k + 1) , (k(7) + (-6))/(k + 1)] = k:1

[(-k - 4)/(k + 1), (7k - 6)/(k + 1)] = k:1

The point lies on the x-axis. i.e.,y = 0

So, 7k - 6/k + 1 = 0

7k - 6 = 0

7k = 6

k = 6/7

Therefore, the ratio of division is 6:7.

To find the coordinates of the point of division,

x coordinates is (m₁x₂ + m₂x₁)/(m₁ + m₂)

Here, m₁:m₂ = 6:7, (x₁ , y₁) = (-4, -6) and (x₂ , y₂) = (-1, 7)

= [6(-1) + 7(-4)]/(6 + 7)

= -6 - 28/13

= -34/13

Therefore, the coordinate of the point of division is (-34/13, 0).

✦ Try This: In what ratio is the line segment joining A(2, -3) and B(5, 6) divide by the x-axis? Also, find the coordinates of the pint of division.

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

NCERT Exemplar Class 10 Maths Exercise 7.3 Problem 10

Summary:

The x–axis divides the line segment joining the points (– 4, – 6) and (–1, 7) in the ratio 6:7. The coordinates of the point of division is (-34/13, 0)

☛ Related Questions:

Question 30 Section Formula Exercise 11

Answer:

Solution:

In coordinate geometry, the Section formula is used to determine the internal or external ratio at which a line segment is divided by a point.

Let the ratio that the point (5,4) divide the line segment joining the points (2,1) and (7,6) be m:n,

Here x1 = 2 , y1 = 1 , x2 = 7, y2 = 6, x = 5, y = 4

By section formula,

$x=\frac{\left(mx_2+nx_1\right)}{(m+n)}\\5=\frac{\left(m\times7+n\times2\right)}{(m+n)}\\5=\frac{\left(7m+2n\right)}{(m+n)}\\5(m+n)=7m+2n\\ 5m+5n=7m+2n\\ 5m-7m=2n-5n\\-2m=-3n\\ \frac{m}{n}=\frac{-3}{-2}=\frac{3}{2}$

Hence the ratio m:n is 3:2.

Was This helpful?

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`