Let $\mathrm{z}, \mathrm{w}$ be complex numbers such that $\overline{\mathrm{z}}+\mathrm{i} \overline{\mathrm{w}}=0$ and arg $\mathrm{zw}=\pi$. Then arg z equals

JEE Main · 2004 · 24 Apr 2004 · Offline · Complex Numbers

Let

z, w

be complex numbers such that

\overline{z} + i \overline{w} = 0

and arg

zw = π

. Then arg z equals

Answer: (C)

Start from

\overset{z}{ˉ} + i \overset{w}{ˉ} = 0

, which gives

\overset{z}{ˉ} = - i \overset{w}{ˉ}

; taking conjugates of both sides turns this into

z = i w

. Then

z w = i w \cdot w = i w^{2}

, so

ar g (z w) = ar g i + 2 ar g w = \frac{π}{2} + 2 ar g w

. Setting this equal to

π

gives

2 ar g w = \frac{π}{2}

, so

ar g w = \frac{π}{4}

. Finally, since

z = i w

ar g z = ar g i + ar g w = \frac{π}{2} + \frac{π}{4} = \frac{3 π}{4}

JEE Main · 2004 · 24 Apr 2004 · Offline · Single correct option