A sphere of radius $3 r$ passes through the origin and meets the coordinate axes in A, B, C then the locus of centroid of $Δ$ ABC is

x^{2} + y^{2} + z^{2} = 4 r^{2}

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

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

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x^{2} + y^{2} + z^{2} = 3 r^{2}

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Solution

The correct option is A $x^{2} + y^{2} + z^{2} = 4 r^{2}$
Equation of sphere OABC is $x^{2} + y^{2} + z^{2} - a x - b y - c z = 0$
where, $\sqrt{\frac{a^{2} + b^{2} + c^{2}}{4}} = 3 r$
$⟹ a^{2} + b^{2} + c^{2} = 9 r^{2}$ ....(i)
From $(i), α^{2} + β^{2} + γ^{2} = 4 r^{2}$
$∴$ Locus is $x^{2} + y^{2} + z^{2} = 4 r^{2}$

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