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Q.

Suppose two complex numbers z = a + ib;w = c + id satisfy the equation $\frac{z + w}{z} = \frac{w}{z + w}$ then

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a

at least one of b & d is non-zero

b

both a & c are zeros

c

both b & d must be non zeros

d

both b & d are zeros

answer is D.

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Detailed Solution

$(z + w)^{2} = zw \Rightarrow z^{2} + zw + w^{2} = 0$
Let $\frac{z}{w} = t \Rightarrow \frac{z}{w} = \frac{- 1 \pm \sqrt{3 i}}{2} \arg z - \arg w = \pm \frac{2 π}{3}$
$\tan^{- 1} \frac{b}{a} - \tan^{- 1} \frac{d}{c} = \pm \frac{2 π}{3} \Rightarrow \frac{bc - ad}{ac + bd} = \pm \sqrt{3}$

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