Find the coordinates of the points which divide internally and externally the line joining $\left(-4, 4\right)$ and $(1,7)$ in the ratio $2 : 1$.
Let $A = \left(-4,4\right)$ and $B = \left(1,7\right)$
Let $P \left(p,q\right)$ divide the line joining $AB$ internally in the ratio $2:1$.
Then, by section formula,
$p = \dfrac{1 \times 2 + \left(-4\right) \times 1}{2 + 1} = \dfrac{2 - 4}{3} = \dfrac{-2}{3}$
$q = \dfrac{7 \times 2 + 4 \times 1}{2 + 1} = \dfrac{14 + 4}{3} = \dfrac{18}{3} = 6$
$\therefore \;$ $P = \left(\dfrac{-2}{3}, 6\right)$
Let $Q \left(x,y\right)$ divide the line joining $AB$ externally in the ratio $2:1$.
Then, by section formula,
$x = \dfrac{1 \times 2 - \left(-4\right) \times 1}{2 - 1} = \dfrac{2 + 4}{1} = 6$
$y = \dfrac{7 \times 2 - 4 \times 1}{2 - 1} = \dfrac{14 - 4}{1} = 10$
$\therefore \;$ $Q = \left(6, 10\right)$