Question

If the equation $a|z{|}^2+\overline{\overline{\alpha }z+\alpha \overline{z}}+d=0$ represents a circle where $a,d$ are real constants, then which of the following condition is correct?

• A. $|\alpha {|}^2-ad\ne 0$
• B. $|\alpha {|}^2-ad>0$ and $a\in R-\left\{0\right\}$
• C. $\alpha =0,a,d\in R^{+}$
• D. $|\alpha {|}^2-ad\ge 0$ and $a\in R$

Answer 1

The correct option is B $|\alpha {|}^2-ad>0$ and $a\in R-\left\{0\right\}$

$a|z{|}^2+\overline{\overline{\alpha }z+\alpha \overline{z}}+d=0$
$\Rightarrow a|z{|}^2+\alpha \overline{z}+\overline{\alpha }z+d=0$
$\Rightarrow z\overline{z}+\left(\frac{\alpha }{a}\right)\overline{z}+\left(\frac{\overline{\alpha }}{a}\right)z+\frac{d}{a}=0$
Centre $=-\frac{\alpha }{a}$
$r=\sqrt{{\left|\frac{\alpha }{a}\right|}^2-\frac{d}{a}}$
$\Rightarrow {\left|\frac{\alpha }{a}\right|}^2\ge \frac{d}{a}$
$\Rightarrow |\alpha {|}^2\ge ad$