Complex Numbers

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]