### Complex Numbers

goals
The closest distance of the origin from a curve given as $a\bar z + \bar az + a\bar a = 0$ ($a$ is a complex number) is.
Consider the equation $az^2+z+1=0$ having purely imaginary root where $a=\cos\theta+i \sin\theta, i=\sqrt {-1}$ and function $f(x)=x^3-3x^2+3(1+\cos\theta)x+5$, then answer the following questions.
Consider the equation $az^2+z+1=0$ having purely imaginary root where $a=\cos\theta+i \sin\theta, i=\sqrt {-1}$ and function $f(x)=x^3-3x^2+3(1+\cos\theta)x+5$, then answer the following questions.
Consider a complex number $z$ which satisfies the equation $\left |z-\left (\displaystyle\frac{4}{z}\right )\right |=2$
Consider a complex number $z$ which satisfies the equation $\left |z-\left (\displaystyle\frac{4}{z}\right )\right |=2$