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

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
  1. $\left(-\frac{2}{3}, \frac{1}{3}, 0\right)$
  2. $(6,-9,12)$
  3. $\left(-\frac{7}{3}, \frac{8}{3},-3\right)$
  4. $\left(\frac{7}{3},-\frac{8}{3}, \frac{9}{3}\right)$

Solution

Let point $P(-9,12,-15)$ divides the line joining $A(1,-2,3)$ and $B(-4,5,-6)$ in ratio $\lambda: 1$. Then, $\quad-9=\frac{-4 \lambda+1}{\lambda+1}, \lambda=-2$ 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) $

Asked in: AP EAMCET 2018 (23 Apr Shift 2)

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