$P \left(3,7\right)$ is a point on the line joining $A \left(1,1\right)$ and $B \left(6,16\right)$. Find the harmonic conjugate of $P$ with respect to $A$ and $B$.
Let the point $P \left(3,7\right)$ divide the line joining $A \left(1,1\right)$ and $B \left(6,16\right)$ internally in the ratio $k : 1$.
Then, by section formula,
$3 = \dfrac{1 \times 1 + 6 \times k}{k + 1}$
i.e. $\;$ $3k + 3 = 1 + 6k$
i.e. $\;$ $3k = 2$ $\implies$ $k = \dfrac{2}{3}$
i.e. $\;$ the point $P$ divides the line $AB$ internally in the ratio $2:3$
Let $Q \left(x,y\right)$ be the harmonic conjugate of $P$ with respect to $A$ and $B$.
Then the point $Q$ divides $AB$ externally in the ratio $2 : 3$
Then, by section formula for external division,
$x = \dfrac{2 \times 6 - 3 \times 1}{2 - 3} = \dfrac{12 - 3}{-1} = -9$
$y = \dfrac{2 \times 16 - 3 \times 1}{2 - 3} = \dfrac{32 - 3}{-1} = -29$
$\therefore \;$ $Q \left(x, y\right) = \left(-9, -29\right)$