Question

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

• A. $(\frac{18}{5},-5,\frac{4}{5})$
  
• B. $(11,-16,2)$
• C. $\left(\frac{1}{2},\frac{1}{3},\frac{1}{4}\right)$
• D. $(0,0,0)$

Answer 1

The correct option is A $(\frac{18}{5},-5,\frac{4}{5})$

Let us assume that point $(2,3,4)$ divides $(3,-2,2)$ and $(6,-17,-4)$ in the ratio $\lambda :1$

We have $\frac{6\lambda +3}{1+\lambda }=2$

$\Rightarrow \lambda =-\frac{1}{4}$

Therefore point $(2,3,4)$ divides given two points in the ratio $1:4$ externally

Let the point which divides the given two points in the ratio $1:4$ internally be $(x,y,z)$

We have $x=\frac{6+12}{5}=\frac{18}{5}$ , $y=\frac{-17-8}{5}=-5$ and $z=\frac{-4+8}{5}=\frac{4}{5}$

Therefore option $A$ is correct