Q179 / 270 · NDA 2021 (Phase 2) · Mathematics

If $z=x+i y$, where $i=\sqrt{-1}$, then what does the equation $z \bar{z}+|z|^2+4(z+\bar{z})-48=0$ represent?

  • A
    Straight line
  • B
    Parabola
  • Correct answer
    C
    Circle
  • D
    Pair of straight lines

Solution

Here $z=x+i y \Rightarrow \bar{z}=x-i y$ $\begin{aligned} \therefore & z \bar{z}+|z|^2+4(z+\bar{z})-48=0 \\ & \Rightarrow|z|^2+|z|^2+4(z+\bar{z})-48=0\left[\therefore z \bar{z}=|z|^2\right] \\ & \Rightarrow 2\left(x^2+y^2\right)+4(2 x)-48=0 \\ & \Rightarrow x^2+y^2+4 x-48=0 \end{aligned}$ Represent equation of circles.

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

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