Question

If for z=α+iβ, |z+2|=z+4(1+i), then α +β and αβ are the roots of the equation

• A. x²+3x-4=0
• B. x²+7x+12=0
• C. x²+2x-3=0
• D. x²+x-12=0

Hint:

For complex equations, separate into real and imaginary parts to solve systematically.

Answer 1

The Correct Option is B

Step 1: Solve the given condition.
\[ |z + 2| = z + 4(1 + i). \] - Substitute \(z = \alpha + i\beta\): \[ |\alpha + i\beta + 2| = (\alpha + 4) + i(\beta + 4). \] This represents a complex number equation where \(z\) is expressed as \( \alpha + i\beta \). 
Step 2: Solve for \(\alpha\) and \(\beta\).
- Expand the magnitudes: \[ \sqrt{(\alpha + 2)^2 + \beta^2} = \sqrt{(\alpha + 4)^2 + (\beta + 4)^2}. \] Now, equate the real and imaginary parts of both sides to solve for \( \alpha \) and \( \beta \): - After solving, we find: \[ \alpha = 1, \quad \beta = -4. \] Step 3: Find the quadratic equation.
- The roots are \( \alpha + \beta = -3 \) and \( \alpha \beta = -4 \). Thus, the quadratic equation is: \[ x^2 - (\alpha + \beta)x + \alpha\beta = x^2 + 7x + 12. \] Final Answer: The quadratic equation is \( x^2 + 7x + 12 = 0 \).