The given equation zz+(4−3i)z+(4+3i)z+c=0 is analyzed to determine the real values of c that form a circle. Key Steps Identifying the Circle's Center: zz is the modulus squared of z, and the terms (4−3i)z and (4+3i)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: √(4)2+(−3)2−c=√25−c The radius needs to be non-negative for the equation to represent a circle: √25−c≥0 which simplifies to: 25−c≥0‌‌⇒‌‌c≤25 Conclusion The real values of c for which the equation represents a circle are in the interval: (−∞,25]