Q48 / 160 · AP EAMCET 2018 (22 Apr Shift 2) · Mathematics

If a point $P$ denotes a complex number $z=x+i y$ in the argand plane and if $\frac{z+1}{z+i}$ is a purely real number, then the locus of $P$ is

  • A
    $x+y+1=0$
  • B
    $x^2+y^2+x+y=0$
  • C
    $x^2+y^2+2 y+1=0,(x, y) \neq(0,-1)$
  • Correct answer
    D
    $x+y+1=0,(x, y) \neq(0,-1)$

Solution

Since $$ \begin{aligned} & \frac{z+1}{z+i}=\frac{x+i y+1}{x+i y+i}=\frac{(x+1)}{x+i(y+1)} \\ & \times \frac{x-i(y+1)}{x-i(y+1)} \end{aligned} $$ If $\frac{z+1}{z+i}$ is a purely real number, $z \neq-i$ $\begin{aligned} & \text { Then, } \quad I_m\left(\frac{z+1}{z+i}\right)=0, z=-i \\ & \Rightarrow \quad x y-(x+1)(y+1)=0, \quad(x, y) \neq(0,-1) \\ & \Rightarrow \quad x y-x y-x-y-1=0 \\ & \Rightarrow \quad x+y+1=0, \quad(x, y) \neq(0,-1)\end{aligned}$

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

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