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(D) The general equation of a circle in the complex plane is given by $z \bar{z} + a \bar{z} + \bar{a} z + b = 0$,where $a$ is a complex constant and

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$(-\infty, 25]$

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(D) The general equation of a circle in the complex plane is given by $z \bar{z} + a \bar{z} + \bar{a} z + b = 0$,where $a$ is a complex constant and $b$ is a real constant. The centre of this circle is $-a$ and the radius is $\sqrt{|a|^2 - b}$. Comparing the given equation $z \bar{z} + (4 - 3i) \bar{z} + (4 + 3i) z + c = 0$ with the general form,we have $a = 4 - 3i$ and $b = c$. For the equation to represent a circle,the radius must be a real number greater than $0$,so $|a|^2 - b > 0$. Here,$|a|^2 = |4 - 3i|^2 = 4^2 + (-3)^2 = 16 + 9 = 25$. Thus,the condition for the radius is $25 - c > 0$,which implies $c If the radius is allowed to be $0$ (a point circle),then $25 - c \geq 0$,which implies $c \leq 25$. Therefore,the set of all real values of $c$ is $(-\infty, 25]$.

The set of all real values of $c$ for which the equation $z \bar{z} + (4 - 3i) \bar{z} + (4 + 3i) z + c = 0$ represents a circle is

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