If ' a ' and ' b ' are complex numbers. One of the roots of the equation x 2 + a x + b = 0 is purely real and the other is purely imaginary then a 2 − a ˉ 2 = kb , find k

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Let α and i β , α , β ∈ R are roots of x 2 + a x + b = 0 ⇒ α + i β = − a , i α β = b α − i β = − a ˉ ⇒ 2 α = − ( a + a ˉ ) and 2 i β = − ( a − a ˉ ) ∴ 4ia β = a 2 − a ˉ 2 ⇒ 4 b = a 2 − a ˉ 2

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