Q48 / 160 · AP EAMCET 2019 (20 Apr Shift 2) · Mathematics

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
    a straight line
  • Correct answer
    B
    a circle
  • C
    a parabola
  • D
    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.

Hence, the correct answer is option: 'B'.

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