# In what ratio does the point (−4, 6) divide the line segment

Question:

In what ratio does the point (−4, 6) divide the line segment joining the points A(−6, 10) and B(3,−8)?

Solution:

The co-ordinates of a point which divided two points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ internally in the ratio $m: n$ is given by the formula,

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

Here it is said that the point (−4,6) divides the points A(−6,10) and B(3,−8). Substituting these values in the above formula we have,

$(-4,6)=\left(\left(\frac{m(3)+n(-6)}{m+n}\right),\left(\frac{m(-8)+n(10)}{m+n}\right)\right)$

Equating the individual components we have,

$-4=\frac{m(3)+n(-6)}{m+n}$

$-4 m-4 n=3 m-6 n$

$7 m=2 n$

$\frac{m}{n}=\frac{2}{7}$

Therefore the ratio in which the line is divided is $2: 7$