The set of all real values of $c$ for which equation $z \bar{z}+(4-3 i) \bar{z}+(4+3 i) z+c=0$ represents a…
The set of all real values of $c$ for which equation $z \bar{z}+(4-3 i) \bar{z}+(4+3 i) z+c=0$ represents a circle is
$[25, \infty)$
$[-5,5]$
$(-\infty,-5] \cup[5, \infty)$
$(-\infty, 25]$
Solution
Since, $z \bar{z}+(4-3 i) \bar{z}+(4+3 i) z+c=0$ represents a circle. Now, general equation of circle is
$z \bar{z}+a \bar{z}+\bar{a} z+b=0$
where, centre $=-a$ and radius $=\sqrt{|a|^2-b}$
So, centre of given circle $=-4+3 i$
$\begin{aligned} & \text { and radius }=\sqrt{|-4+3 i|^2-C}=\sqrt{5^2-C}=\sqrt{25-C} \\ & \text { For existence of circle, radius } \geq 0 \Rightarrow \sqrt{25-C} \geq 0 \\ & \Rightarrow 25-C \geq 0 \Rightarrow 25 \geq C \Rightarrow C=(-\infty, 25]\end{aligned}$