Q48 / 160 · TS EAMCET 2020 (14 Sep Shift 2) · Mathematics

If the amplitude of $z-2-3 i$ is $\pi / 4$, then the locus of $z=x+i y$ is

  • A
    $x+y-1=0$
  • B
    $x-y-1=0$
  • C
    $x+y+1=0$
  • Correct answer
    D
    $x-y+1=0$

Solution

$\begin{aligned} & \text { Given, } \arg (z-2-3 i)=\frac{\pi}{4} \Rightarrow z=x+i y \\ & z-2-3 i=x+i y-2-3 i=(x-2)+(y-3) i \\ & \arg (z-2-3 i)=\tan ^{-1}\left(\frac{y-3}{x-2}\right)=\frac{\pi}{4} \Rightarrow \frac{y-3}{x-2}=\tan \frac{\pi}{4} \\ & \quad y-3=x-2 \Rightarrow x-y+1=0 \end{aligned}$ $\therefore$ Locus of $z$ is $x-y+1=0$.

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

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