Question

A circle whose radius is $r$ and centre $Z_0$, then the equation of the circle is

• A. $Z\overline{Z}-Z{\overline{Z}}_0-\overline{Z}{Z}_0+Z_0{\overline{Z}}_0=r^2$
• B. $Z\overline{Z}+Z{\overline{Z}}_0-\overline{Z}{Z}_0+Z_0{\overline{Z}}_0=r^2$
• C. $Z\overline{Z}-Z{\overline{Z}}_0+{\overline{Z}}_0-Z_0{\overline{Z}}_0=r^2$
• D. none of these

Answer 1

The correct option is A $Z\overline{Z}-Z{\overline{Z}}_0-\overline{Z}{Z}_0+Z_0{\overline{Z}}_0=r^2$

Given, radius of a circle is $r$ and centre is $Z_0$.
Equation of circle is $|Z-Z_0|=r$
$\Rightarrow |Z-Z_0{|}^2=r^2$
$\Rightarrow (Z-Z_0)(\overline{Z}-\overline{Z_0})=r^2$
$\Rightarrow Z\overline{Z}-Z{\overline{Z}}_0-\overline{Z}{Z}_0+Z_0{\overline{Z}}_0=r^2$
Hence, option A.