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(A) Step $1$: Distance of point $(1, 2, 3)$ from origin $(0, 0, 0)$ is $d_1 = \sqrt{(1-0)^2 + (2-0)^2 + (3-0)^2} = \sqrt{1 + 4 + 9} = \sqrt{14}$.Step

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$8$

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(A) Step $1$: Distance of point $(1, 2, 3)$ from origin $(0, 0, 0)$ is $d_1 = \sqrt{(1-0)^2 + (2-0)^2 + (3-0)^2} = \sqrt{1 + 4 + 9} = \sqrt{14}$.Step $2$: Distance of point $(1, 2, 3)$ from plane $2x - 3y + z + k = 0$ is $d_2 = \frac{|2(1) - 3(2) + 1(3) + k|}{\sqrt{2^2 + (-3)^2 + 1^2}} = \frac{|2 - 6 + 3 + k|}{\sqrt{4 + 9 + 1}} = \frac{|k - 1|}{\sqrt{14}}$.Step $3$: Given product $d_1 	imes d_2 = 7$, so $\sqrt{14} 	imes \frac{|k - 1|}{\sqrt{14}} = 7$.Step $4$: $|k - 1| = 7$, which implies $k - 1 = 7$ or $k - 1 = -7$.Step $5$: Thus, $k = 8$ or $k = -6$. Given the options, $k = 8$ is the correct value.

If the product of the distances of the point $(1, 2, 3)$ from the origin and the plane $2x - 3y + z + k = 0$ is $7$, then the value of $k$ is

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