Complex Numbers
Unimodular Complex Numbers
Complex Numbers_PYQ
Grade 11
Question:
A complex number $z$ is said to be unimodular if $|z|=1$. If $z_1$ and $z_2$ are complex numbers such that $\dfrac{z_1-2z_2}{2-z_1\bar{z}_2}$ is unimodular and $z_2$ is not unimodular, then the point $z_1$ lies on a
straight line parallel to $X$-axis
straight line parallel to $Y$-axis
circle of radius $2$
circle of radius $\sqrt{2}$
Step-by-Step Solution
Key Concept: Expanding $|w|=1$ and collecting terms yields the factored form $(|z_1|^2-4)(1-|z_2|^2)=0$. The given constraint on $z_2$ selects the factor about $z_1$.
**Step 1: Set up the modulus condition**
$\left|\dfrac{z_1-2z_2}{2-z_1\bar{z}_2}\right|=1 \Rightarrow |z_1-2z_2|^2=|2-z_1\bar{z}_2|^2$.
**Step 2: Expand both sides**
LHS: $|z_1|^2-2z_1\bar{z}_2-2\bar{z}_1 z_2+4|z_2|^2$. RHS: $4-2\bar{z}_1 z_2-2z_1\bar{z}_2+|z_1|^2|z_2|^2$.
**Step 3: Cancel and factor**
$|z_1|^2+4|z_2|^2=4+|z_1|^2|z_2|^2 \Rightarrow (|z_1|^2-4)(1-|z_2|^2)=0$.
**Step 4: Apply the constraint**
Since $z_2$ is not unimodular, $|z_2|\neq1$, so $|z_1|^2=4$, i.e., $z_1$ lies on a circle of radius $2$.
Correct Answer: 3