Find the ratio in which the line segment joining $(-2, -3)$ and $(5, 6)$ is divided by y-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 y-axis.

To do:

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

Solution:

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

This implies,

Its abscissa is $0$.

Let the point $(0, y)$ 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,

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

\( \Rightarrow \frac{5 m-2 n}{m+n}=0 \)

\( \Rightarrow 5 m-2 n=0 \)

\( \Rightarrow 5 m=2 n \)

\( \Rightarrow \frac{m}{n}=\frac{2}{5} \)

\( \Rightarrow m:n=2:5 \)

This implies, \( y=\frac{2(6)+5(-3)}{2+5} \)

\( =\frac{12-15}{7} \)

\( =\frac{-3}{7} \)

The ratio of the division is $2:5$ and the coordinates of the point of division are \( (0,\frac{-3}{7}) \).

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Updated on: 10-Oct-2022

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