fourth quadrant
We are given the complex numbers: \[ Z_1 = \sqrt{3} + i\sqrt{3}, \quad Z_2 = \sqrt{3} + i. \]
Step 1: Compute \( \frac{Z_1}{Z_2} \)
Dividing \( Z_1 \) by \( Z_2 \): \[ \frac{Z_1}{Z_2} = \frac{\sqrt{3} + i\sqrt{3}}{\sqrt{3} + i}. \] Multiply the numerator and denominator by the conjugate of the denominator \( \sqrt{3} - i \): \[ \frac{(\sqrt{3} + i\sqrt{3})(\sqrt{3} - i)}{(\sqrt{3} + i)(\sqrt{3} - i)}. \] The denominator simplifies as: \[ (\sqrt{3} + i)(\sqrt{3} - i) = 3 - (-1) = 4. \] Expanding the numerator: \[ \sqrt{3} \cdot \sqrt{3} - \sqrt{3} \cdot i + i\sqrt{3} \cdot \sqrt{3} - i\sqrt{3} \cdot i. \] \[ = 3 - \sqrt{3} i + 3i - \sqrt{3} (-1). \] \[ = 3 - \sqrt{3} i + 3i + \sqrt{3}. \] \[ = (3 + \sqrt{3}) + (-\sqrt{3} + 3)i. \] Dividing by 4: \[ \frac{Z_1}{Z_2} = \frac{3+\sqrt{3}}{4} + i \frac{3-\sqrt{3}}{4}. \]
Step 2: Compute \( \left( \frac{Z_1}{Z_2} \right)^{50 \)}
Since \( \frac{Z_1}{Z_2} \) has a positive real part and a positive imaginary part, raising it to any power will still lie in the first quadrant.
Step 3: Conclusion
Thus, the point \( (x,y) \) lies in: \[ \boxed{\text{First Quadrant}}. \]
If \( x \) and \( y \) are two positive real numbers such that \( x + iy = \frac{13\sqrt{5} + 12i}{(2 - 3i)(3 + 2i)} \), then \( 13y - 26x = \):
\[ \text{The domain of the real-valued function } f(x) = \sin^{-1} \left( \log_2 \left( \frac{x^2}{2} \right) \right) \text{ is} \]
Let \( A = [a_{ij}] \) be a \( 3 \times 3 \) matrix with positive integers as its elements. The elements of \( A \) are such that the sum of all the elements of each row is equal to 6, and \( a_{22} = 2 \).
\[ \textbf{If } | \text{Adj} \ A | = x \text{ and } | \text{Adj} \ B | = y, \text{ then } \left( | \text{Adj}(AB) | \right)^{-1} \text{ is } \]