Determine the ratio in which the point $\left(\dfrac{-2}{5}, x\right)$ divides the joining of $\left(-4,3\right)$ and $\left(2,8\right)$. Also find the value of $x$.
Let the point $P \left(\dfrac{-2}{5}, x\right)$ divide the line join of $A \left(-4,3\right)$ and $B \left(2,8\right)$ in the ratio $k : 1$.
Then by section formula,
$\dfrac{-2}{5} = \dfrac{2k - 4}{k + 1}$
i.e. $\;$ $-2k -2 = 10 k - 20$
i.e. $\;$ $12 k = 18$ $\implies$ $k = \dfrac{18}{12} = \dfrac{3}{2}$
i.e. $\;$ the point $P$ divides $AB$ in the ratio $3 : 2$.
Again, by section formula
$x = \dfrac{3 \times 8 + 2 \times 3}{3 + 2}$
i.e. $\;$ $x = \dfrac{24 + 6}{5} = 6$
$\therefore \;$ $P = \left(\dfrac{-2}{5}, 6\right)$