Complex Numbers
Modulus Condition Leading to Quadratic
nta_pyq_2023_apr
Grade 11
Question:
If for $z=\alpha+i\beta$, $|z+2|=z+4(1+i)$, then $\alpha+\beta$ and $\alpha\beta$ are the roots of the equation
x^2+3x-4=0
x^2+7x+12=0
x^2+x-12=0
x^2+2x-3=0
Step-by-Step Solution
Key Concept: Equate real and imaginary parts: $\sqrt{(\alpha+2)^2+\beta^2}=\alpha+4$ (real) and $\beta+4=0$ (imaginary).
$\beta=-4,\ \alpha=1$. Sum of roots $=-3$, product $=-4$. Equation: $x^2+7x+12=0$.
Correct Answer: 2