JEE Advanced 2022 Paper 1 · Q04 · Complex Numbers

Question

Let $z$ be a complex number with non-zero imaginary part. If $\dfrac{2+3z+4z^{2}}{2-3z+4z^{2}}$ is a real number, then the value of $|z|^{2}$ is _____________.

SolveKar AI · Accepted Answer

Let $w=\dfrac{2+3z+4z^{2}}{2-3z+4z^{2}}$. Then $w=1+\dfrac{6z}{2-3z+4z^{2}}$, so $w\in\mathbb{R}\iff \dfrac{z}{2-3z+4z^{2}}\in\mathbb{R}\iff \dfrac{1}{4z+	frac{2}{z}-3}\in\mathbb{R}\iff 4z+\dfrac{2}{z}-3\in\mathbb{R}$. The number $4z+2/z$ is real $\iff \operatorname{Im}\!\left(4z+\dfrac{2}{z}ight)=0$. Writing $z=x+iy$, $\operatorname{Im}(4z)=4y$ and $\operatorname{Im}(2/z)=\operatorname{Im}\!\left(\dfrac{2\bar{z}}{|z|^{2}}ight)=\dfrac{-2y}{|z|^{2}}$. So $4y-\dfrac{2y}{|z|^{2}}=0$. Since $y
eq 0$, $|z|^{2}=1/2=0.50$.

JEE Advanced 2022 Paper 1 · Q04 · Complex Numbers

Solution