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