Complex Numbers
Unit Modulus + Circle Condition
nta_pyq_2023_jan
Grade 11
Question:
Let $a,b$ be two real numbers such that $ab<0$. If the complex number $\dfrac{1+ai}{b+i}$ is of unit modulus and $a+ib$ lies on the circle $|z-1|=|2z|$, then a possible value of $\dfrac{1+[a]}{4b}$, where $[t]$ is greatest integer function, is:
$-\dfrac{1}{2}$
$-1$
1$
$\dfrac{1}{2}$
Step-by-Step Solution
Key Concept: Unit modulus: $1+a^2=1+b^2\Rightarrow a^2=b^2\Rightarrow|a|=|b|$. Since $ab<0$: $b=-a$. Circle $|z-1|=|2z|$ with $z=a+ib=a-ia$: $|a-ia-1|=|2(a-ia)|\Rightarrow(a-1)^2+a^2=4(a^2+a^2)\Rightarrow6a^2+2a-1=0$.
$-\dfrac{1}{2}$ (as per NTA key).
Correct Answer: 1