Question

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

Answer 1

Let $(5,1)$ divides the line joining $P$ & $Q$

in $\lambda :1$

so, $X-$coordinate of $R=\frac{6\lambda +2}{\lambda +1}$

$Y-$coordinate of $R=\frac{-2\lambda +10}{\lambda +1}$

$\Rightarrow \frac{6\lambda +2}{\lambda +1}=5$

$\Rightarrow 6\lambda +2=5\lambda +5$

$\lambda =3$

so, we need to find a point which divide $P$ and $Q$ in $3:1$

$\therefore R_X=\frac{(-3)\left(6\right)+2}{-3+1}=-16/-2=8$

$R_y=\frac{(-3)(-2)+10}{-3+1}=16/-2=-8$

$\therefore$ harmonic conjugate of

$(5,1)$ is $(8,-8)$