If $x^{2} - (a + b) x + a b = 0,$ then the value of ${(x - a)}^{2} + {(x - b)}^{2}$ is

a^{2} + b^{2}

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{(a + b)}^{2}

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{(a - b)}^{2}

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a^{2} - b^{2}

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Solution

The correct option is C ${(a - b)}^{2}$
$x^{2} - x (a + b) + a b = 0 \Rightarrow x^{2} - a x - b x = - a b$
$(x - a)^{2} + (x - b)^{2} = x^{2} + a^{2} - 2 a x + x^{2} + b^{2} - 2 x b = 2 x^{2} - 2 a x - 2 b x + a^{2} + b^{2} = 2 (x^{2} - a x - b x) + a^{2} + b^{2} = - 2 a b + a^{2} + b^{2} = (a - b)^{2}$

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