Find the ratio in which the y-axis divides the line segment joining the points (5, −6)
Question:

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.

Solution:

The ratio in which the $\mathrm{y}$-axis divides two points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ is $\lambda: 1$

The co-ordinates of the point dividing two points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ in the ratio $m: n$ is given as,

$(x, y)=\left(\left(\frac{\lambda x_{2}+x_{1}}{\lambda+1}\right),\left(\frac{\lambda y_{2}+y_{1}}{\lambda+1}\right)\right)$ where, $\lambda=\frac{m}{n}$

Here the two given points are A(5,−6) and B(−1,−4).

$(x, y)=\left(\frac{-\lambda+5}{\lambda+1}, \frac{-4 \lambda-6}{\lambda+1}\right)$

Since, the y-axis divided the given line, so the x coordinate will be 0.

$\frac{-\lambda+5}{\lambda+1}=0$

$\lambda=\frac{5}{1}$

Thus the given points are divided by the $y$-axis in the ratio $5: 1$.

The co-ordinates of this point (x, y) can be found by using the earlier mentioned formula.

$(x, y)=\left(\left(\frac{\frac{5}{1}(-1)+(5)}{\frac{5}{1}+1}\right) \cdot\left(\frac{\frac{5}{1}(-4)+(-6)}{\frac{5}{1}+1}\right)\right)$

$(x, y)=\left(\left(\frac{0}{6}\right),\left(-\frac{26}{6}\right)\right)$

$(x, y)=\left(0,-\frac{26}{6}\right)$

Thus the co-ordinates of the point which divides the given points in the required ratio are $\left(0,-\frac{26}{6}\right)$.