Calculations for the principal argument $\text{Arg}(Z) \in (-\pi, \pi]$:

• Multiply the two complex numbers:
  $$Z_1 Z_2 = (-2 + 2i)(3i) = -6i + 6i^2 = -6 - 6i$$
• Notice that for $Z = -6 - 6i$, both real and imaginary parts are negative, placing it in the third quadrant.
• The principal argument in the third quadrant is:
  $$\text{Arg}(Z_1 Z_2) = -\pi + \arctan\left(\frac{-6}{-6}\right) = -\pi + \frac{\pi}{4} = -\frac{3\pi}{4}$$

In coordinate geometry of the standard hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$:
The two vertices are located at $A(a, 0)$ and $A'(-a, 0)$.
The segment joining the two vertices is called the transverse axis, and its length is $2a$, which is the distance between the two vertices.
The distance between the two foci is $2ae$ (focal distance).

To find the center, first divide the equation of the circle by 4 so the leading coefficients are 1:

Compare this with the general form of the circle equation $x^2 + y^2 + 2gx + 2fy + c = 0$:
$2g = -2 \implies g = -1$
$2f = 3 \implies f = \frac{3}{2}$

The center of the circle is given by $(-g, -f)$: