Question

The harmonic conjugate of $(2,3,4)$ w.r.t. the points $(3,-2,2)$ and $(6,-17,-4)$ is?

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

Answer 1

Answer: (D)

$(\frac{18}{5},-5,\frac{4}{5})$

Let A be $(3,-2,2)$ and B be $(6,-17,-4)$

C be $(2,3,4)$

Let harmonic conjugate be $D(a,b,c)$

$\frac{AC}{CB}=\frac{-AD}{DB}$, if D is harmonic conjugate of C.

Let $\frac{AC}{CB}=\frac{m}{n}$

$\frac{3n+6m}{m+n}=2=>4m=-n\frac{m}{n}=\frac{-1}{4}\frac{AD}{DB}=\frac{-AC}{CB}=\frac{1}{4}D=(a,b,c)=(\frac{3\times 4+6\times 1}{4+1},\frac{-2\times 4+(-17)\times 1}{5},\frac{2\times 4-4\times 1}{5})D=(\frac{18}{5},-5,\frac{4}{5})$