Question

If \( Z_1 \) and \( Z_2 \) are roots of equation \( Z^2 + 4Z - (1 + 12i) = 0 \), where \( Z \) is complex number, then the value of \( |Z_1|^2 + |Z_2|^2 \) is:

• A. 34
• B. 37
• C. 42
• D. 45

Hint:

To find the square root of a complex number \( a+ib \), the real part of the root is \( \pm \sqrt{\frac{|Z|+a}{2}} \) and the imaginary part is \( \pm \sqrt{\frac{|Z|-a}{2}} \).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
For a quadratic equation \( aZ^2 + bZ + c = 0 \), the sum of roots is \( -b/a \) and the product of roots is \( c/a \). We use these relations along with the property \( |Z|^2 = Z \cdot \bar{Z} \).

Step 2: Key Formula or Approach:
1. \( Z_1 + Z_2 = -4 \). 2. \( Z_1 Z_2 = -(1 + 12i) \). 3. \( |Z_1|^2 + |Z_2|^2 \) can be found by solving for \( Z_1 \) and \( Z_2 \) specifically using the quadratic formula.

Step 3: Detailed Explanation:
1. Quadratic Formula: \( Z = \frac{-4 \pm \sqrt{16 + 4(1 + 12i)}}{2} = \frac{-4 \pm \sqrt{20 + 48i}}{2} = -2 \pm \sqrt{5 + 12i} \). 2. To find \( \sqrt{5 + 12i} \), let \( (x+iy)^2 = 5 + 12i \): \[ x^2 - y^2 = 5, \quad 2xy = 12 \implies xy = 6 \] Solving gives \( x=3, y=2 \) (since \( 3^2 - 2^2 = 5 \)). 3. Roots: \[ Z_1 = -2 + (3 + 2i) = 1 + 2i \] \[ Z_2 = -2 - (3 + 2i) = -5 - 2i \] 4. Calculate moduli squared: \[ |Z_1|^2 = 1^2 + 2^2 = 5 \] \[ |Z_2|^2 = (-5)^2 + (-2)^2 = 25 + 4 = 29 \] 5. Sum: \( 5 + 29 = 34 \).

Step 4: Final Answer:
The value is 34.