Find the ratio in which the line $x = 0$ divides line $AB$ where $A = \left(-4,4\right)$ and $B = \left(2, 8\right)$. Also find the coordinates of the point where $AB$ is divided by the Y axis.
Given: $\;$ $A \left(x_1, y_1\right) = \left(-4,4\right)$; $\;$ $B \left(x_2, y_2\right) = \left(2, 8\right)$
Any point on the line $x = 0$ (i.e. the Y axis) is $P \left(0,p\right)$
Let point $P$ divide line $AB$ in the ratio $k : 1$
Then by section formula, $\;$ $0 = \dfrac{2k - 4}{k + 1}$
i.e. $\;$ $2k - 4 = 0$
i.e. $\;$ $2k = 4$ $\implies$ $k = 2$
$\therefore \;$ Any point on the line $x = 0$ divides the line $AB$ in the ratio $2 : 1$
Now, by section formula, $\;$ $p = \dfrac{2 \times 8 + 1 \times 4}{2 + 1} = \dfrac{16 + 4}{3} = \dfrac{20}{3}$
$\therefore \;$ The coordinates of the point where $AB$ is divided by the Y axis is $\left(0, \dfrac{20}{3}\right)$