Let z, w be complex numbers such that z¯ +i w =0 and Ag (z w)=π then Ag(z)=

z¯ +i w =0 ⇒z−iw=0⇒z=iw⇒w=−izArgzw=π ⇒arg⁡−iz2=π ⇒−π2+2arg⁡(z)=π Arg(−i)=−π2⇒2arg⁡(z)=3π2 ∴arg⁡(z)=3π4

Let z, w be complex numbers such that z¯ +i w =0 and Ag (z w)=π then Ag(z)=

z¯ +i w =0 ⇒z−iw=0⇒z=iw⇒w=−izArgzw=π ⇒arg⁡−iz2=π ⇒−π2+2arg⁡(z)=π Arg(−i)=−π2⇒2arg⁡(z)=3π2 ∴arg⁡(z)=3π4

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Let z, w be complex numbers such that $\bar{z} + i w = 0$ and Ag (z w)= $π$ then Ag(z)=

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$\frac{3 π}{4}$

$\frac{π}{2}$

$\frac{5 π}{4}$

$\frac{π}{4}$

answer is C.

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Detailed Solution

$\bar{z} + i w = 0 \begin{aligned} \Rightarrow z - i w = 0 \\ \Rightarrow z = i w \\ \Rightarrow w = - i z \end{aligned}$

$A r g (z w) = π$ $\Rightarrow \arg (- {iz}^{2}) = π$

$\begin{array}{l} \Rightarrow - \frac{π}{2} + 2 \arg (z) = π (A r g (- i) = - \frac{π}{2}) \\ \Rightarrow 2 \arg (z) = \frac{3 π}{2} \\ ∴ \arg (z) = \frac{3 π}{4} \end{array}$

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Let z, w be complex numbers such that $\bar{z} + i w = 0$ and Ag (z w)= $π$ then Ag(z)=