Find the ratio in which the y-axis divides the line segment joining the points $(5, -6)$ and $(-1, -4)$. Also, find the coordinates of the point of division.

Given:

The line segment joining the points $(5, -6)$ and $(-1, -4)$ is divided by the y-axis.

To do:

We have to find the ratio and 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 $(5, -6)$ and $(-1, -4)$ 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,

\( (0, y)=\left(\frac{m \times (-1)+n \times (5)}{m+n}, \frac{m \times (-4)+n \times(-6)}{(m+n)}\right) \)

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

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

\( \Rightarrow m=5 n \)

\( \Rightarrow \frac{m}{n}=\frac{5}{1} \)

\( \Rightarrow m:n=5:1 \)

This implies, \( y=\frac{5(-4)+1(-6)}{5+1} \)

\( =\frac{-20-6}{6} \)

\( =\frac{-26}{6} \)

\( =\frac{-13}{3} \) 

The ratio of division is $5:1$ and the coordinates of the point of division are \( (0,\frac{-13}{3}) \).

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Updated on: 10-Oct-2022

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