Question

The harmonic conjugate of $(2,3,4)$ with respect to the points $(3,-2,2),(6,-17,-4)$ is

• A. $\left(\frac{1}{2},\frac{1}{3},\frac{1}{4}\right)$
• B. $\left(\frac{18}{5},-5,\frac{4}{5}\right)$
• C. $\left(\frac{-18}{5},\frac{5}{4},\frac{4}{5}\right)$
• D. $\left(\frac{18}{5},-5,\frac{-4}{5}\right)$

Answer 1

Answer: (B)

Let $P(2,3,4),A(3,-2,2)$ and $B(6,-17,-4)$.
Let $P$ divides $AB$ in the ratio $k:1$, then
$2=\frac{6k+3}{k+1}$
$\Rightarrow 2k+2=6k+3$
$\Rightarrow -4k=1$
$\Rightarrow k=-\frac{1}{4}$
Harmonic conjugate $Q$ divides in the ratio $-k:1$, i.e. $\frac{1}{4}:1$
$Q=\left[\frac{\frac{1}{4}(6)+3}{\frac{1}{4}+1},\frac{\frac{1}{4}(-17)-2}{\frac{1}{4}+1},\frac{\frac{1}{4}(-4)+2}{\frac{1}{4}+1}\right]$
$=\left(\frac{6+12}{5},\frac{-17-8}{5},\frac{-4-8}{5}\right)$
$=\left(\frac{18}{5},-\frac{25}{5},\frac{4}{5}\right)$
$=\left(\frac{18}{5},-5,\frac{4}{5}\right)$