If the roots of the equation $x^{4} - 10 x^{3} + 50 x^{2} - 130 x + 169 = 0$ are of the form a ± ib and b ± ia then (a, b) =

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(–3, –2)

(3, 2)

(–3, 2)

(2, 1)

answer is A.

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Detailed Solution

$\begin{array}{l} I f a \pm i b, b \pm i a a r e r o o t s o f x^{4} - 10 x^{3} + 50 x^{2} - 130 x + 169 = 0 \\ ∴ a + i b + a - i b + b + i a + b - i a = 10 \\ \Rightarrow 2 (a + b) = 10 \Rightarrow a + b = 5 - - (1) \\ ∴ (a + i b) (a - i b) (b + i a) (b - i a) = 169 \\ \Rightarrow (a^{2} + b^{2}) (b^{2} + a^{2}) = 169 \\ \Rightarrow (a^{2} + b^{2}) = 169 \\ \Rightarrow a^{2} + b^{2} = 13 \\ \Rightarrow {(a + b)}^{2} - 2 a b = 13 \\ \Rightarrow 25 - 2 a b = 13 \\ \Rightarrow 2 a b = 12 \Rightarrow a b = 6 \\ ∴ a - b = \sqrt{{(a + b)}^{2} - 4 a b} = \sqrt{25 - 24} = 1 - - (2) \\ (1) + (2) \Rightarrow 2 a = 6 \Rightarrow a = 3 \\ (1) - (2) \Rightarrow 2 b = 4 \Rightarrow b = 2 \end{array}$