The equation of the sphere through the points $(1, 0, 0), (0, 1, 0)$ and $(1, 1, 1)$ and having the smallest radius is

3 (x^{2} + y^{2} + z^{2}) - 4 x - 4 y - 2 z + 1 = 0

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2 (x^{2} + y^{2} + z^{2}) - 3 x - 2 y - z + 1 = 0

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x^{2} + y^{2} + z^{2} - x - y + z + 1 = 0

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x^{2} + y^{2} + z^{2} - 2 x - 2 y + 4 z + 1 = 0

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Solution

The correct option is B $3 (x^{2} + y^{2} + z^{2}) - 4 x - 4 y - 2 z + 1 = 0$
Given, $(1, 0, 0), (0, 1, 0)$ and $(1, 1, 1)$

The triangle formed by these three points is equilateral.

So, centroid is the centre and radius is equal to circumradius.

$R =$ circumradius $= \frac{a}{\sqrt{3}} = \frac{1}{\sqrt{3}}$
Centre $= (\frac{2}{3}, \frac{2}{3}, \frac{1}{3})$

So, the equation is $3 (x^{2} + y^{2} + z^{2}) - 4 x - 4 y - 2 z + 1 = 0$

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