If $a - b + c = 5$ and $c a - a b - b c = 7,$ then value of $a^{2} + b^{2} + c^{2}$ is

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Solution

Given:
$a - b + c = 5$ and $c a - a b - b c = 7$
We know that,
$(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 (a b + b c + c a)$
Replacing $b \to - b$ , we get
$\Rightarrow (a - b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 (- a b - b c + c a) \Rightarrow 25 = a^{2} + b^{2} + c^{2} + 2 \times 7 ∴ a^{2} + b^{2} + c^{2} = 11$

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