Question

The equation of a circle passing through the origin and having intercepts of $a$ and $b$ on real and imaginary axis respectively on the argand plane is given by

• A. $z\overline{z}=aIm\left(z\right)+bRe\left(z\right)$
• B. $z\overline{z}=aRe\left(z\right)+bIm\left(z\right)$
• C. $z\overline{z}=aRe\left(z\right)-bIm\left(z\right)$
• D. $z\overline{z}=aIm\left(z\right)-bRe\left(z\right)$

Answer 1

The correct option is B $z\overline{z}=aRe\left(z\right)+bIm\left(z\right)$


From figure,
$\arg \left(\frac{z-a}{z-ib}\right)=\frac{\pi }{2}$
$\Rightarrow \frac{z-a}{z-ib}+\frac{\overline{z}-a}{\overline{z}+ib}=0$
$\Rightarrow z\overline{z}-a\left(\frac{z+\overline{z}}{2}\right)-b\left(\frac{z-\overline{z}}{2i}\right)=0$
$\Rightarrow z\overline{z}=aRe\left(z\right)+bIm\left(z\right)$