Complex Numbers
Modulus Condition on Complex Numbers
nta_pyq_2024_apr
Grade 11
Question:
If $z_1$, $z_2$ are two distinct complex numbers such that $\left|\dfrac{z_1-2z_2}{\frac{1}{2}-z_1\bar{z}_2}\right|=2$, then:
$z_1$ lies on a circle of radius $\frac{1}{2}$ and $z_2$ lies on a circle of radius 1.
both $z_1$ and $z_2$ lie on the same circle.
either $z_1$ lies on a circle of radius $\frac{1}{2}$ or $z_2$ lies on a circle of radius 1.
either $z_1$ lies on a circle of radius 1 or $z_2$ lies on a circle of radius $\frac{1}{2}$.
Step-by-Step Solution
Key Concept: Square both sides and simplify: $(|z_1|^2-1)(|2z_2|^2-1)=0\Rightarrow(|z_1|^2-1)(4|z_2|^2-1)=0$.
$(|z_1|^2-1)(4|z_2|^2-1)=0\Rightarrow|z_1|=1$ or $|z_2|=1/2$.
Correct Answer: 4