Let z and w be complex numbers such that [ z + i w = 0] and arg zw = [ ].

Question

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

Accepted Answer

Given $$\overline z  + i\overline w  = 0$$

$$ \Rightarrow \overline z  =  - i\overline w $$

$$ \Rightarrow \overline{\overline z}  =  - \overline {i\overline w } $$

$$ \Rightarrow \overline{\overline z}  =  - \overline i \overline{\overline w} $$

$$ \Rightarrow z =  - \overline i w$$

$$ \Rightarrow z =  - \left( { - i} \right)w$$

$$ \Rightarrow z = iw$$

Now given that Arg(zw) = $$\pi $$

$$ \Rightarrow $$ Arg(z$$ \times $$ ${z \over i}$) =$\pi $$ \Rightarrow $Arg(z2) - Arg(i) =$\pi $$ \Rightarrow $2Arg(z) -${\pi  \over 2}$=$\pi $$

[ $$i$$ complex number represent (0, 1) point on imaginary axis and Arg($$i$$) means the angle  made by the point (0, 1) with real axis which is $${\pi  \over 2}$$]

$$ \Rightarrow $2Arg(z) =${{3\pi } \over 2}$$ \Rightarrow $Arg(z) =${{3\pi } \over 4}$

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