Question

The harmonic conjugate of the point $C(-6,-2)$ with respect to the points $A(-10,0)$ and $B(0,-5)$ is

• A. $(20,10)$
• B. $(10,30)$
• C. $(10,-20)$
• D. $(-30,10)$

Answer 1

The correct option is D $(-30,10)$

Given : $A(-10,0),B(0,-5)$
Let point $C$ divides the line segment $AB$ in the ratio of $k:1$ internally. Then
$C\equiv (\frac{-10}{k+1},\frac{-5k}{k+1})\Rightarrow (-6,-2)=(\frac{-10}{k+1},\frac{-5k}{k+1})\Rightarrow k=\frac{2}{3}$

Let $P$ be the harmonic conjugate of point $C$, so it divides line segment $AB$ in ratio of $2:3$ externally. Then
$P\equiv (\frac{30}{2-3},\frac{-10}{2-3})\therefore P\equiv (-30,10)$