The equation of the sphere concentric with the sphere $x^{2} + y^{2} + z^{2} - 4 x - 2 y - 6 z - 7 = 0$ and passing through $(0, 0, 0)$ is :

x^{2} + y^{2} + z^{2} - 4 x - 2 y - 6 z = 0

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

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

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None of these

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Solution

The correct option is A $x^{2} + y^{2} + z^{2} - 4 x - 2 y - 6 z = 0$
Let the required sphere be
$x^{2} + y^{2} + z^{2} + 2 g x + 2 f y + 2 w z + d = 0$ ...(i)
Now it is concentric withh
$x^{2} + y^{2} + z^{2} - 4 x - 2 y = 6 z - 7 = 0$
$∴ g = - 2, f = - 1, w = - 3$
Also (i) passes through $(0, 0, 0)$ we get $d = 0$
$∴$ required sphere is
$x^{2} + y^{2} + z^{2} - 4 x - 2 y - 6 z = 0$

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