Question

The harmonic conjugate of $P(-9,12,-15)$ with respect to the line segment $AB$, where $A=(1,-2,3)$ and $B=(-4,5,-6)$ is

• A. $\left(-\frac{2}{3},\frac{1}{3},0\right)$
• B. $(6,-9,12)$
• C. $\left(-\frac{7}{3},\frac{8}{3},-3\right)$
• D. $\left(\frac{7}{3},-\frac{8}{3},\frac{9}{3}\right)$

Answer 1

Answer: (C)

Let point $P(-9,12,-15)$ divides the line joining
$A(1,-2,3)$ and $B(-4,5,-6)$ in ratio $\lambda :1$
Then, $-9=\frac{-4\lambda +1}{\lambda +1},\lambda =-2$
So, harmonic conjugate of point ${}^{\prime }{P}^{\prime }$ with respect to the line segment $AB$ will divides the line segment $AB$ internally in ratio $2:1$, so point will be
$\left(\frac{-8+1}{3},\frac{10-2}{3},\frac{-12+3}{3}\right)$
$=\left(-\frac{7}{3},\frac{8}{3},-3\right)$