If the equation


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

  1. (1) $|\alpha|^{2}-\operatorname{ad} \neq 0$

  2. (2) $|\alpha|^{2}-a d>0$ and $a \in R-\{0\}$

  3. (3) $|\alpha|^{2}-a d \geq 0$ and $a \in R$

  4. (4) $\alpha=0, \mathrm{a}, \mathrm{d} \in \mathrm{R}^{+}$

Correct Option: , 2


$\mathrm{az} \overline{\mathrm{z}}+\alpha \overline{\mathrm{z}}+\bar{\alpha} \mathrm{z}+\mathrm{d}=0 \rightarrow$ Circle

centre $=\frac{-\alpha}{\mathrm{a}} \quad 2=\sqrt{\frac{\alpha \bar{\alpha}}{\mathrm{a}^{2}}-\frac{\mathrm{d}}{\mathrm{a}}}=\sqrt{\frac{\alpha \bar{\alpha}-\mathrm{ad}}{\mathrm{a}^{2}}}$

So $|\alpha|^{2}-\mathrm{ad}>0 \& \mathrm{a} \in \mathrm{R}-\{0\}$

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