The harmonic conjugate of (2,3,4) with respec | vedclass

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(D) Let $A = (3, -2, 2)$ and $B = (6, -17, -4)$. Let $P = (2, 3, 4)$ divide the line segment $AB$ in the ratio $m:n$. Using the section formula,we hav

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

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(D) Let $A = (3, -2, 2)$ and $B = (6, -17, -4)$. Let $P = (2, 3, 4)$ divide the line segment $AB$ in the ratio $m:n$. Using the section formula,we have: $P = \left(\frac{6m + 3n}{m+n}, \frac{-17m - 2n}{m+n}, \frac{-4m + 2n}{m+n}\right) = (2, 3, 4)$. Equating the $x$-coordinates: $\frac{6m + 3n}{m+n} = 2 \implies 6m + 3n = 2m + 2n \implies 4m = -n \implies \frac{m}{n} = -\frac{1}{4}$. Thus,$P$ divides $AB$ externally in the ratio $1:4$. The harmonic conjugate of $P$ with respect to $A$ and $B$ divides the segment $AB$ internally in the ratio $m:n = 1:4$. Using the section formula for internal division: $Q = \left(\frac{1(6) + 4(3)}{1+4}, \frac{1(-17) + 4(-2)}{1+4}, \frac{1(-4) + 4(2)}{1+4}\right) = \left(\frac{6+12}{5}, \frac{-17-8}{5}, \frac{-4+8}{5}\right) = \left(\frac{18}{5}, -5, \frac{4}{5}\right)$. Therefore,the correct option is $D$.

The harmonic conjugate of $(2,3,4)$ with respect to the points $(3,-2,2)$ and $(6,-17,-4)$ is

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