In what ratio is the line segment joining the points $(-2, -3)$ and $(3, 7)$ divided by the y-axis? Also find the coordinates of the point of division.

Given:

The line segment joining the points $(-2, -3)$ and $(3, 7)$ is divided by the y-axis.

To do:

We have to find 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 $(3, 7)$ 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,

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

\( \Rightarrow \frac{3 m-2 n}{m+n}=0 \)

\( \Rightarrow 3 m-2 n=0 \)
\( \Rightarrow 3 m=2 n \)

\( \Rightarrow \frac{m}{n}=\frac{2}{3} \)
\( \Rightarrow m:n=2:3 \)

This implies,
\( y=\frac{2(7)+3(-3)}{2+3} \)

\( =\frac{14-9}{5} \)

\( =\frac{5}{5} \)

\( =1 \)
The coordinates of the point of division are \( (0,1) \).

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Updated on: 10-Oct-2022

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