Q51 / 160 · AP EAMCET 2020 (18 Sep Shift 2) · Mathematics

The equation of any in the complex plane is of the form \(z \bar{z}+b \bar{z}+\bar{b} z+c=0\) where \((b \in \mathbf{C}, c \in \mathbf{R})\)

  • Correct answer
    A
    Circle
  • B
    Straight line
  • C
    Parabola
  • D
    Hyperbola

Solution

Let Equation of circle in cartesian system be \(x^2+y^2+2 g x+2 f y+c=0\) ...(i) \(\begin{aligned} \text{Let } z & =x+i y \\ \bar{z} & =x-i y \\ z+\bar{z} & =2 x \\ z-\bar{z} & =2 i y \\ 2 y & =\frac{z-\bar{z}}{i}=i(\bar{z}-z) \\ z \cdot \bar{z} & =(x+i y)(x-i y) \\ z \cdot \bar{z} & =x^2+y^2 \end{aligned}\) \(\therefore\) From Eq. (i) \(\begin{aligned} z \bar{z}+g(z+\bar{z})+i(\bar{Z}-Z) f+c & =0 \\ z \bar{z}+(g-i f) z+(g+i f) \bar{z}+c & =0 \\ z \bar{z}+\bar{b} z+b \bar{z}+c & =0 \quad \text { [Let } b=g+\text { if] } \end{aligned}\) Hence, option (a) is correct.

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

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