Q193 / 270 · NDA 2015 (Phase 2) · Mathematics

$z \bar{z}+(3-i) z+(3+i) \bar{z}+1=0$ represents a circle with

  • Correct answer
    A
    centre $(-3,-1)$ and radius 3
  • B
    centre $(-3,1)$ and radius 3
  • C
    centre $(-3,-1)$ and radius 4
  • D
    centre $(-3,1)$ and radius 4

Solution

$\begin{aligned} & \text { Let } \mathrm{z}=\mathrm{x}+\mathrm{iy} \\ & \overline{\mathrm{z}}=\mathrm{x}-\mathrm{iy} \\ & \Rightarrow \overline{\mathrm{z}}+(3-\mathrm{i}) \mathrm{z}+(3+\mathrm{i}) \overline{\mathrm{z}}+1=0 \\ & \Rightarrow(\mathrm{x}+\mathrm{iy})(\mathrm{x}-\mathrm{iy})+(3-\mathrm{i})(\mathrm{x}+\mathrm{iy})+(3+\mathrm{i})(\mathrm{x}-\mathrm{iy})+1=0 \\ & \Rightarrow \mathrm{x}^2+\mathrm{y}^2+6 \mathrm{x}+2 \mathrm{y}+1=0 \\ & \Rightarrow(\mathrm{x}+3)^2-9+(\mathrm{y}+1)^2-1+1=0 \\ & \Rightarrow(\mathrm{x}+3)^2+(\mathrm{y}+1)^2=(3)^2 \\ & \text {Centre }(-3,-1) \\ & \text {radius }=3\end{aligned}$

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

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