Find the coordinates of the point which divides the line segment joining the points $(4, - 3)$ and $(8, 5)$ in the ratio $3 : 1$ internally.

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Solution

Let $P (x, y)$ be the required point. Using the section formula
$\begin{matrix} P (x, y) = (\frac{m_{1} x_{2} + m_{2} x_{1}}{m_{1} + m_{2}}, \frac{m_{1} y_{2} + m_{2} y_{1}}{m_{1} + m_{2}}), we get x = \frac{3 (8) + 1 (4)}{3 + 1} = \frac{24 + 4}{4} = \frac{28}{4} = 7, y = \frac{3 (5) + 1 (- 3)}{3 + 1} = \frac{15 - 3}{4} = \frac{12}{4} = 3 \end{matrix}$
$P (x, y) = (7, 3)$ is the required point.

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