Question

The harmonic conjugate of $P(-9,12,-15)$ with respect to the line segment $A B$, 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 $'P'$ with respect to the line segment $A B$ will divides the line segment $A B$ internally in ratio $2: 1$, so point will be
$\left(\frac{-8+1}{3}, \frac{10-2}{3}, \frac{-12+3}{3}\right)$
$=\left(-\frac{7}{3}, \frac{8}{3},-3\right)$