Complex Numbers

The given equation $z \overline{z}+(4-3i) z+(4+3i) \overline{z}+c=0$ is analyzed to determine the real values of $c$ that form a circle.Key StepsIdentifying the Circle's Center:$z \overline{z}$ is the modulus squared of $z$, and the terms $(4-3i) z$ and $(4+3i) \overline{z}$ suggest the transformation involves the complex number $4-3i$.Thus, the center of the circle is at $4-3i$.Finding the Radius:To find the radius, use the formula that incorporates the modulus of the center:$\sqrt{(4)^2+(-3)^2-c}=\sqrt{25-c}$The radius needs to be non-negative for the equation to represent a circle:$\sqrt{25-c} \geq 0$which simplifies to:$25-c \geq 0 \;\; \Rightarrow \;\; c \leq 25$ConclusionThe real values of $c$ for which the equation represents a circle are in the interval: $(-\infty, 25]$