If $a^{2} + b^{2} + c^{2} = 20, a + b + c = 0, f i n d a b + b c + c a .$

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Solution

$a^{2} + b^{2} + c^{2} = 20, a + b + c = 0 ∴ (a + b + c)^{2} = 0 a^{2} + b^{2} + c^{2} + 2 (a b + b c + c a) = 0 \Rightarrow 20 + 2 (a b + b c + c a) = 0 \Rightarrow 2 (a b + b c + c a) = - 20 ∴ a b + b c + c a = \frac{- 20}{2} = - 10$

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