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(C) Let the point be $P(x, y, z)$. The given condition is $PF_1 + PF_2 = 10$,where $F_1 = (4, 0, 0)$ and $F_2 = (-4, 0, 0)$.$\sqrt{(x-4)^2 + y^2 + z^2

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$9x^2 + 25y^2 + 25z^2 = 225$

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(C) Let the point be $P(x, y, z)$. The given condition is $PF_1 + PF_2 = 10$,where $F_1 = (4, 0, 0)$ and $F_2 = (-4, 0, 0)$.$\sqrt{(x-4)^2 + y^2 + z^2} + \sqrt{(x+4)^2 + y^2 + z^2} = 10$Let $S = x^2 + y^2 + z^2 + 16$. Then the equation is $\sqrt{S - 8x} + \sqrt{S + 8x} = 10$.Squaring both sides: $(S - 8x) + (S + 8x) + 2\sqrt{S^2 - 64x^2} = 100$$2S + 2\sqrt{S^2 - 64x^2} = 100 \Rightarrow S + \sqrt{S^2 - 64x^2} = 50$$\sqrt{S^2 - 64x^2} = 50 - S$Squaring again: $S^2 - 64x^2 = 2500 - 100S + S^2$$100S - 64x^2 = 2500$Substitute $S = x^2 + y^2 + z^2 + 16$:$100(x^2 + y^2 + z^2 + 16) - 64x^2 = 2500$$100x^2 + 100y^2 + 100z^2 + 1600 - 64x^2 = 2500$$36x^2 + 100y^2 + 100z^2 = 900$Dividing by $4$: $9x^2 + 25y^2 + 25z^2 = 225$.

$A$ point moves such that the sum of its distances from the points $(4, 0, 0)$ and $(-4, 0, 0)$ remains $10$. The locus of the point is

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