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Algebra of Complex Numbers

If ax +by + c...

Question

If ax +by + cz = p, then minimum value of $x^{2} + y^{2} + z^{2}$ is

{(\frac{p}{a + b + c})}^{2}

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\frac{p^{2}}{a^{2} + b^{2} + c^{2}}

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\frac{a^{2} + b^{2} + c^{2}}{p^{2}}

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{(\frac{a + b + c}{p})}^{2}

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Solution

The correct option is C $\frac{p^{2}}{a^{2} + b^{2} + c^{2}}$
Given $a x + b y + c z = p$
$(a x + b y + c z)^{2} \leq (a^{2} + b^{2} + c^{2}) (x^{2} + y^{2} + z^{2}) ⟹ x^{2} + y^{2} + z^{2} \geq (\frac{a x + b y + c z}{a^{2} + b^{2} + c^{2}})^{2}$
$⟹ x^{2} + y^{2} + z^{2} \geq (\frac{p^{2}}{a^{2} + b^{2} + c^{2}})$

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\in R^{+}

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