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  4. If z=x+i y satisfies amp (z-1)= textamp(z+3 i) then the value of (x-1): y is equal to

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Q. If $z=x+i y$ satisfies amp $(z-1)=\text{amp}(z+3 i)$ then the value of $(x-1): y$ is equal to

Complex Numbers and Quadratic Equations

Solution:

We have, $\text{amp}(z-1)=\text{amp}(z+3 i)$
$\Rightarrow \tan ^{-1} \frac{y}{x-1}=\tan ^{-1} \frac{y+3}{x} $
$\Rightarrow x y=(x-1)(y+3)$
$\Rightarrow 3(x-1)=y$
$\therefore (x-1): y=1: 3 .$