If A and B are (1, 4) and (5, 2) respectively,
Question:

If A and B are (1, 4) and (5, 2) respectively, find the coordinates of P when AP/BP = 3/4.

Solution:

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(1,4) and B(5,2). Let point P(x, y) divide the line joining ‘AB’ in the ratio

Substituting these values in the earlier mentioned formula we have,

$(x, y)=\left(\left(\frac{\frac{3}{4}(5)+(1)}{\frac{3}{4}+1}\right),\left(\frac{\frac{3}{4}(2)+(4)}{\frac{3}{4}+1}\right)\right)$

$\left.(x, y)=\left(\frac{\frac{15+4(1)}{4}}{\frac{3+4}{4}}\right),\left(\frac{\frac{6+4(4)}{4}}{\frac{3+4}{4}}\right)\right)$

$(x, y)=\left(\left(\frac{19}{7}\right),\left(\frac{22}{7}\right)\right)$

Thus the co-ordinates of the point which divides the given points in the required ratio are $\left(\frac{19}{7}, \frac{22}{7}\right)$.