In what ratio does the x axis divides the line segment joining the points ( - 4,6) and (--1, 7)

In what ratio does the x axis divide the line segment joining the points 4 6 and 1,7 find the coordinates of the point of division

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Text Solution

Solution : Let the required ratio be `lambda :1`. So , the coordinates of the point M of division A (-4,-6) and B(-1,7) are <br> `{(lambda x_(2)+1*x_(1))/(lambda+1),(lambda y_(2)+1*y_(1))/(lambda+1)}` <br> Here ,`x_(1)=-4,x_(2)=-1 and y_(1)=-6,y_(2)=7` <br> i.e., `((lambda(-1)+1(-4))/(lambda+1),(lambda(7)+1*(-6))/(lambda+1))=((-lambda-4)/(lambda+1),(7lambda-6)/(lambda+1))` <br> But according to the question , line segment joining A(-4,-6) and B(-1,7) is divided by the X - axis . So , y- coordinate must be zero. <br> `:. (7lambda-6)/(lambda+1)rArr7lambda-6=0` <br> `:. lambda=(6)/(7)` <br> So , the required ratio is `6 : 7` and the point of division M is `{(-(6)/(7)-4)/((6)/(7)+1),(7xx(6)/(7)-6)/((6)/(7)+1)}` <br> i.e., `((-34/(17))/(13/(7)),(6-6)/(13/(7)))` i.e., `((-34)/(13),0)`. <br> Hence , the required point of division is `((-34)/(13),0)`.

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In what ratio does the x-axis divide the line segment joining the points (-4, -6) and (-1, 7). Also, find the coordinates of the point of division.

Asked by Topperlearning User | 04 Jun, 2014, 01:23: PM

Let the line segment joining the points (-4, -6) and (-1, 7) be divided by the point on x-axis (x, 0) in the ratio k: 1.

Using section formula, we have:

Thus, the coordinates of the point of division are

Answered by | 04 Jun, 2014, 03:23: PM

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)

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