Q. The equation of a circle whose radius and centre are $r\ and\ \mathcal{z}_{0}$ respectively, is
A) $\mathcal{z}\overline{\mathcal{z}}\mathcal{- z}{\overline{\mathcal{z}}}_{0} - \overline{\mathcal{z}}\mathcal{z}_{0} + \mathcal{z}_{0}{\overline{\mathcal{z}}}_{0} = r^{2}$
B) $\mathcal{z}\overline{\mathcal{z}}\mathcal{+ z}{\overline{\mathcal{z}}}_{0} - \overline{\mathcal{z}}\mathcal{z}_{0} + \mathcal{z}_{0}{\overline{\mathcal{z}}}_{0} = r^{2}$
C) $\mathcal{z}\overline{\mathcal{z}}\mathcal{- z}{\overline{\mathcal{z}}}_{0} + \overline{\mathcal{z}}\mathcal{z}_{0} - \mathcal{z}_{0}{\overline{\mathcal{z}}}_{0} = r^{2}$
D) None of the above
Explanation
(a)
Equation of circle whose centre is $\mathcal{z}_{0}$ and radius is $r$, is $\left| \mathcal{z -}\mathcal{z}_{0} \right|^{2} = r^{2}$
$\Rightarrow \left( \mathcal{z -}\mathcal{z}_{0} \right)(\overline{\mathcal{z -}\mathcal{z}_{0})} = r^{2}$
$\Rightarrow \ \left( \mathcal{z -}\mathcal{z}_{0} \right)\left( \overline{\mathcal{z}} - {\overline{\mathcal{z}}}_{0} \right) = r^{2}$
$\mathcal{\Rightarrow \ z}\overline{\mathcal{z}}\mathcal{- z}{\overline{\mathcal{z}}}_{0} - \overline{\mathcal{z}}\mathcal{z}_{0} + \mathcal{z}_{0}{\overline{\mathcal{z}}}_{0} = r^{2}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.