JEE Main 2023 (Online) 10th April Evening ShiftQuestion 1
Let $S=\left\{z=x+iy:\frac{2z-3i}{4z+2i}\text{ is a real number}\right\}$. Then which of the following is NOT correct?
Complex Numbers
Solution
$\frac{2z-3i}{4z+2i}=\frac12\cdot\frac{z-\frac32 i}{z+\frac12 i}$ real (with $z\ne-\frac i2$). Imag part of $(x+i(y-\frac32))(x-i(y+\frac12))$ is $-2x=0\Rightarrow x=0$, $y\ne-\frac12$. So A, B hold; C: with $x=0$, $y+y^2=-\frac14$ only at $y=-\frac12$ (excluded), so C holds. D, the point $(0,-\frac12)$, is excluded, so NOT correct. Option D.