Question

Find the harmonic conjugates of the point $R(5,1)$ with respect to the points $P(2,10)$ and $Q(6,-2)$

Answer 1

R.E.F image

$\frac{6k+2}{k+1}=5$

$\frac{-2k+10}{k+1}=\frac{-6+10}{4}$
$=\frac{4}{4}-1$

$6k+2=2k+5$

$k=3$

$3:1$ externally

$\frac{3Q-P}{3-1}$

Harmonic conjugate $\Rightarrow \frac{3Q-P}{3-1}$

$\Rightarrow \frac{3\left(6\right)-\left(2\right)}{3-1}$,$\frac{3(-2)-1\left(10\right)}{3-1}$

$\Rightarrow \frac{18-2}{2},\frac{-6-10}{2}$

$\Rightarrow (8,-8)$