Complex Numbers
Real-Valued Condition on Complex Fraction — Locus
nta_pyq_2023_apr
Grade 11
Question:
Let $S=\left\{z=x+iy:\ \dfrac{2z-3i}{4z+2i}\text{ is a real number}\right\}$. Then which of the following is NOT correct?
$y+x^2+y^2\neq-\dfrac{1}{4}$
$(x,y)=\left(0,-\dfrac{1}{2}\right)$
$x=0$
$y\in\left(-\infty,-\dfrac{1}{2}\right)\cup\left(-\dfrac{1}{2},\infty\right)$
Step-by-Step Solution
Key Concept: Set imaginary part of $\frac{2z-3i}{4z+2i}$ equal to zero. After simplification, this gives $x=0$ with $y\neq-\frac{1}{2}$.
$x=0$, $y\neq-\frac{1}{2}$. Option (2) claims $(0,-\frac{1}{2})\in S$, which is false.
Correct Answer: 2