In the complex plane $C$, the set $\left\{z \in C: \arg \left(\frac{z-1}{z+1}\right)=\frac{\pi}{4}\right\}$…
In the complex plane $C$, the set $\left\{z \in C: \arg \left(\frac{z-1}{z+1}\right)=\frac{\pi}{4}\right\}$ represents
- a straight line
- a circle
- a parabola
- an ellipse
Solution
Let a complex number $z=x+i y$,
then
$
\begin{aligned}
\frac{z-1}{z+1} & =\frac{(x-1)+i y}{(x+1)+i y} \times \frac{(x+1)-i y}{(x+1)-i y} \\
& =\frac{\left(x^2+y^2-1\right)+i y(x+1-x+1)}{(x+1)^2+y^2} \\
& =\frac{\left(x^2+y^2-1\right)}{(x+1)^2+y^2}+i \frac{2 y}{(x+1)^2+y^2}
\end{aligned}
$
According to the given information,
$
\begin{gathered}
\arg \left(\frac{z-1}{z+1}\right)=\frac{\pi}{4} \Rightarrow \frac{2 y}{\left(x^2+y^2-1\right)}=1 \\
\Rightarrow \quad x^2+y^2-2 y-1=0
\end{gathered}
$
$\therefore$ The complex plane $C$ represents a circle.
Hence, option (b) is correct
Asked in: AP EAMCET 2019 (20 Apr Shift 2)
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