Discuss the nature of the roots of the equation:
$2 (a^{2} + b^{2}) x^{2} + 2 (a + b) x + 1 = 0$

Real and equal

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Real and unequal

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Imaginary and in conjugate pair

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None of the above

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Solution

The correct option is D Imaginary and in conjugate pair
$2 (a^{2} + b^{2}) x^{2} + 2 (a + b) x + 1 = 0 D = B^{2} - 4 A C = 4 {(a + b)}^{2} - 4 (2 (a^{2} + b^{2})) = {4 a}^{2} + {4 b}^{2} + 8 a b - 8 a^{2} - {8 b}^{2} = - {4 a}^{2} - {4 b}^{2} + 8 a b = - {(a - b)}^{2} < 0$
Hence, roots are imaginary and in conjugate pair.

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