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:
We solve the quadratic equation \( Z^2 + 4Z - (1 + 12i) = 0 \) using the quadratic formula.
Step 2: Key Formula or Approach:
\[ Z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Step 3: Detailed Explanation:
Here \( a = 1, b = 4, c = -(1 + 12i) \).
\[ Z = \frac{-4 \pm \sqrt{16 + 4(1 + 12i)}}{2} = \frac{-4 \pm \sqrt{16 + 4 + 48i}}{2} = \frac{-4 \pm \sqrt{20 + 48i}}{2} \]
\[ Z = \frac{-4 \pm 2\sqrt{5 + 12i}}{2} = -2 \pm \sqrt{5 + 12i} \]
To find \( \sqrt{5 + 12i} \), let \( \sqrt{5 + 12i} = x + iy \).
\( x^2 - y^2 = 5 \) and \( 2xy = 12 \implies xy = 6 \).
By inspection or solving, \( x = 3, y = 2 \) works (\( 9 - 4 = 5 \)).
So, \( \sqrt{5 + 12i} = \pm(3 + 2i) \).
Roots are:
\( Z_1 = -2 + (3 + 2i) = 1 + 2i \)
\( Z_2 = -2 - (3 + 2i) = -5 - 2i \)
Magnitudes:
\( |Z_1|^2 = 1^2 + 2^2 = 5 \)
\( |Z_2|^2 = (-5)^2 + (-2)^2 = 25 + 4 = 29 \)
Value \( = |Z_1|^2 + |Z_2|^2 = 5 + 29 = 34 \).
Step 4: Final Answer:
The sum of the squares of the magnitudes is 34.