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 a complex number $$z$$ which satisfies the equation $$\left |z-\left (\displaystyle\frac{4}{z}\right )\right |=2$$