Question

If \( Z_1 = \sqrt{3} + i\sqrt{3} \) and \( Z_2 = \sqrt{3} + i \), and \[ \left( \frac{Z_1}{Z_2} \right)^{50} = x + iy, \] then the point \( (x,y) \) lies in:

• A. first quadrant
• B. second quadrant
• C. third quadrant
• D. fourth quadrant

Hint:

To determine the quadrant of a complex number, analyze its real and imaginary parts. If both are positive, the number lies in the first quadrant.

Answer 1

The Correct Option is A

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: \[ \boxed{\text{First Quadrant}}. \]