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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
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The correct answer is:
$\left(-\frac{7}{3}, \frac{8}{3},-3\right)$
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
$$
\left(\frac{-8+1}{3}, \frac{10-2}{3}, \frac{-12+3}{3}\right)=\left(-\frac{7}{3}, \frac{8}{3},-3\right)
$$
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
$$
\left(\frac{-8+1}{3}, \frac{10-2}{3}, \frac{-12+3}{3}\right)=\left(-\frac{7}{3}, \frac{8}{3},-3\right)
$$
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