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  4. The equation of any in the complex plane is of the form z barz+b barz+ barb z+ c=0 where (b ∈ C, c ∈ R)

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

AP EAMCETAP EAMCET 2020

Solution:

Let Equation of circle in cartesian system be
$x^{2}+y^{2}+2 g x+2 f y+ c=0 \ldots$ (i)
Let $Z=x +i y$
$\bar{Z}=x-i y$
$Z+\bar{Z}=2 x$
$Z-\bar{Z}=2 i y$
$2 y=\frac{Z-\bar{Z}}{i}=i(\bar{z}-z)$
$z \cdot \bar{z}=(x +i y)(x-i y)$
$z \cdot \bar{Z}=x^{2}+y^{2}$
$\therefore $ From Eq. (i)
$Z \bar{z}+g(z+\bar{z})+i(\bar{z}-z) f+ c=0$
$z \bar{Z}+(g-i f) Z+(g +i f) \bar{z}+c=0$
$z \bar{Z}+\bar{b} z +b \bar{z}+c=0\,\, [$ Let $b=g+$ if $]$