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Sunday, May 5, 2024

2024/033) If z is complex number and imaginary part of z is non zero and \frac{2+3z-4z^2}{2 - 3z+ 4z^2} \in R then find |z|^2

 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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