Find the ratio in which the line segment joining $(-2, -3)$ and $(5, 6)$ is divided by x-axis. Also, find the co-ordinates of the point of division in each case.

Given:

The line segment joining the points $(-2, -3)$ and $(5, 6)$ is divided by the x-axis.

To do:

We have to find the ratio of division and coordinates of the point of division.

Solution:

The point which divides the given line segment lies on x-axis.

This implies,

Its ordinate is $0$.

Let the point $(x, 0)$ intersects the line segment joining the points $(-2, -3)$ and $(5, 6)$ in the ratio $m : n$.

Using section formula, we have,

\( (x, y)=(\frac{mx_{2}+nx_{1}}{m+n}, \frac{my_{2}+ny_{1}}{m+n}) \)

Therefore,

\( (x, 0)=\left(\frac{m \times 5+n(-2)}{m+n}, \frac{m \times 6+n \times(-3)}{(m+n)}\right) \)

\( \Rightarrow \frac{6 m-3 n}{m+n}=0 \)

\( \Rightarrow 6 m-3 n=0 \)

\( \Rightarrow 6 m=3 n \)

\( \Rightarrow \frac{m}{n}=\frac{3}{6} \)

\( \Rightarrow m:n=1:2 \)

This implies,

\( x=\frac{1(5)+2(-2)}{1+2} \)

\( =\frac{5-4}{3} \)

\( =\frac{1}{3} \) 

The ratio of the division is $1:2$ and the coordinates of the point of division are \( (\frac{1}{3},0) \).

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Updated on: 10-Oct-2022

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