Question

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

• A. Circle
• B. Straight line
• C. Parabola
• D. Hyperbola

Answer 1

Answer: (A)

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