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(C) The distance of the point $P(1, 1, 1)$ from the origin $O(0, 0, 0)$ is given by $d_1 = \sqrt{(1-0)^2 + (1-0)^2 + (1-0)^2} = \sqrt{3}$.The distance

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$4$

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(C) The distance of the point $P(1, 1, 1)$ from the origin $O(0, 0, 0)$ is given by $d_1 = \sqrt{(1-0)^2 + (1-0)^2 + (1-0)^2} = \sqrt{3}$.The distance of the point $P(1, 1, 1)$ from the plane $x - y + z + k = 0$ is given by $d_2 = \frac{|1 - 1 + 1 + k|}{\sqrt{1^2 + (-1)^2 + 1^2}} = \frac{|1 + k|}{\sqrt{3}}$.The product of the distances is $d_1 	imes d_2 = 5$.Substituting the values,we get $\sqrt{3} 	imes \frac{|1 + k|}{\sqrt{3}} = 5$.This simplifies to $|1 + k| = 5$.Case $1$: $1 + k = 5 \implies k = 4$.Case $2$: $1 + k = -5 \implies k = -6$.Comparing with the given options,the correct value is $k = 4$.

If the product of distances of the point $(1, 1, 1)$ from the origin and the plane $x - y + z + k = 0$ is $5$,then $k =$

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