JEE Advanced 2022 Paper 1Question 1
Let $z$ be a complex number with non-zero imaginary part. If $$\frac{2+3z+4z^2}{2-3z+4z^2}$$ is a real number, then the value of $|z|^2$ is __________.
Complex Numbers
Answer$0.50$
Solution
Dividing by $z$, the ratio is $\frac{w+3}{w-3}$ with $w=4z+\frac2z$, real iff $w$ is real. $\operatorname{Im}w=4y-\frac{2y}{|z|^2}=0$ with $y\ne0$ gives $|z|^2=\frac12$.