A plane passes through a fixed point $(a, b, c)$ . The locus of the foot of the perpendicular to it from the origin is a sphere of radius

\sqrt{a^{2} + b^{2} + c^{2}}

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\frac{1}{2} \sqrt{a^{2} + b^{2} + c^{2}}

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a^{2} + b^{2} + c^{2}

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None of these

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Solution

The correct option is B $\frac{1}{2} \sqrt{a^{2} + b^{2} + c^{2}}$
Let the foot of the perpendicular from the origin on the given plane be $P (α, β, γ)$
Since, the plane passes through $A (a, b, c)$ .
$∴ A P ⊥ O P \Rightarrow α (α - a) + β (β - b) + γ (γ - c) = 0$
Hence, the locus of $(α, β, γ)$ is
$x (x - a) + y (y - b) + z (z - c) = 0 \Rightarrow x^{2} + y^{2} + z^{2} - a x - b y - c z = 0$
which is a sphere of radius $= \frac{1}{2} \sqrt{a^{2} + b^{2} + c^{2}}$

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A plane passes through a fixed point (a,b,c). The locus of the foot of the perpendicular to it from the origin is a sphere of radius

A plane passes through a fixed point $(a, b, c)$ . The locus of the foot of the perpendicular to it from the origin is a sphere of radius