If the equation [a|z|^2 + z + z + d = 0] represents a circle where a, d are real constants…

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?

Accepted Answer

$$a|z{|^2} + \alpha \overline z  + \overline \alpha  z + d = 0$$

$$ \Rightarrow $$ $$z\overline z  + \left( {{\alpha  \over a}} \right)\overline z  + \left( {{{\overline \alpha  } \over a}} \right)z + {d \over a} = 0$$

$$ \therefore $$ Centre $$ =  - {\alpha  \over a}$$

$$r = \sqrt {{{\left| {{\alpha  \over a}} \right|}^2} - {d \over a}} $$

$$ \Rightarrow {\left| {{\alpha  \over a}} \right|^2} \ge {d \over a}$$

$$ \Rightarrow {\left| \alpha  \right|^2} \ge ad$$

Solutions · 1