Find the ratio in which the line segment joining the points $A (3, -3)$ and $B (-2, 7)$ is divided by x-axis. Also, find the coordinates of the point of division.

Given:

The line segment joining the points $A(3, -3)$ and $B(-2, 7)$ 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 $P(x, 0)$ intersects the line segment joining the points $A(3, -3)$ and $B(-2, 7)$ 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,

\( P(x, 0)=\left(\frac{m \times (-2)+n \times 3}{m+n}, \frac{m \times 7+n \times(-3)}{(m+n)}\right) \)

\( \Rightarrow \frac{7m-3 n}{m+n}=0 \)

\( \Rightarrow 7m-3 n=0 \)

\( \Rightarrow 7 m=3 n \)

\( \Rightarrow \frac{m}{n}=\frac{3}{7} \)

\( \Rightarrow m:n=3:7 \)

This implies,

\( x=\frac{3(-2)+7(3)}{3+7} \)

\( =\frac{-6+21}{10} \)

\( =\frac{15}{10} \) 

\( =\frac{3}{2} \)

The ratio of the division is $3:7$ and the coordinates of the point of division are \( (\frac{3}{2},0) \).

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Updated on: 10-Oct-2022

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