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Equation of Family of Circles Touching a Line and Passing through a Given Point on the Line

The equation ...

Question

The equation of the sphere which passes through the four points $(0, 0, 0), (1, 0, 0), (0, 1, 0)$ and $(0, 0, 1)$ is

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

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

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

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

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Solution

The correct option is B $x^{2} + y^{2} + z^{2} - x - y - z = 0$
Equation of sphere passes through origin is given by,
$x^{2} + y^{2} + z^{2} + a x + b y + c z = 0 . . . (1)$
Given, it also passes through $(1, 0, 0), (0, 1, 0), 0, 0, 1)$
$1 + a = 0 \Rightarrow a = - 1$
$1 + b = 0 \Rightarrow b = - 1$
and $1 + c = 0 \Rightarrow c = - 1$
Hence, required sphere is
$x^{2} + y^{2} + z^{2} - x - y - z = 0$

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