We have
\frac{2+3z-4z^2}{2 - 3z+ 4z^2} \in R
So \frac{2+3z-4z^2}{2 - 3z+ 4z^2} -1 \in R
Or \frac{6z}{2 - 3z+ 4z^2} \in R
Or \frac{2 - 3z+ 4z^2}{6z} \in R
Or \frac{2 - 3z+ 4z^2}{z} \in R
Or \frac{2}{z} -3 + 4z \in R
Or \frac{2}{z} + 4z \in R
Let z = x + iy and y\ne 0
So we have \frac{2}{x+iy} + 4(x+iy) \in R
Or \frac{2(x-iy)}{x^2+y^2} + 4(x+iy) \in R
Or \frac{-2y}{x^2+y^2} + 4y = 0 as imaginary part has to be zero
So as we have y is not zero dividing by y we get x^2+y^2 = \frac{1}{2} or |z|^2=\frac{1}{2}
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