The equation of the sphere concentric with $x^{2} + y^{2} + z^{2} - 2 x - 4 y - 6 z - 11 = 0$ but double the radius is

x^{2} + y^{2} + z^{2} - 2 x - 4 y - 6 z - 22 = 0

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x^{2} + y^{2} + z^{2} + 2 x + 4 y + 6 z + 22 = 0

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x^{2} + y^{2} + z^{2} + 2 x + 4 y + 6 z - 86 = 0

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x^{2} + y^{2} + z^{2} - 2 x - 4 y - 6 z - 86 = 0

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Solution

The correct option is D $x^{2} + y^{2} + z^{2} - 2 x - 4 y - 6 z - 86 = 0$
Given sphere is,
$x^{2} + y^{2} + z^{2} - 2 x - 4 y - 6 z - 11 = 0$
$\Rightarrow (x - 1)^{2} + (y - 2)^{2} + (z - 3)^{2} = 5^{2}$
Hence, centre and radius of required sphere are $(1, 2, 3)$ and $10$ .
Ergo equation of this sphere is,
$(x - 1)^{2} + (y - 2)^{2} + (z - 3)^{2} = 10^{2}$
$\Rightarrow x^{2} + y^{2} + z^{2} - 2 x - 4 y - 6 z - 86 = 0$

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If the radius of the sphere $x^{2} + y^{2} + z^{2} - 2 x - 4 y - 6 z$ = 0 is r, then find the value of $r^{2}$

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