The point $R \left(22,23\right)$ divides the line join of $P \left(7,5\right)$ and $Q$ externally in the ratio $3 : 5$. Find $Q$.
Let $Q = \left(p,q\right)$
The point $R \left(22,23\right)$ divides the line joining $P \left(7,5\right)$ and $Q \left(p,q\right)$ externally in the ratio $3:5$.
Then, by section formula,
$22 = \dfrac{p \times 3 - 7 \times 5}{3 - 5}$
i.e. $\;$ $22 = \dfrac{3p - 35}{-2}$
i.e. $\;$ $3p = -44 + 35$ $\implies$ $p = \dfrac{-9}{3} = -3$
and $\;$ $23 = \dfrac{q \times 3 - 5 \times 5}{3 - 5}$
i.e. $\;$ $23 = \dfrac{3q - 25}{-2}$
i.e. $\;$ $3q = - 46 + 25$ $\implies$ $q = \dfrac{-21}{3} = -7$
$\therefore \;$ $Q = \left(-3, -7\right)$