Let z be a complex number with non-zero imaginary part. If 2 + 3 z + 4 z 2 2 - 3 z + 4 z 2 is a real number, then the value of | z | 2 is _______. [2022]

Question

Let $z$ be a complex number with non-zero imaginary part. If $\frac{2 + 3 z + 4 z^{2}}{2 - 3 z + 4 z^{2}}$ is a real number, then the value of $| z |^{2}$ is _______. [2022]

Smart Achievers · Accepted Answer

(0.50)

$Let X = \frac{4 z^{2} + 3 z + 2}{4 z^{2} - 3 z + 2}$

$It can be written as = 1 + \frac{6 z}{4 z^{2} - 3 z + 2}; Now X = 1 + \frac{6}{2 (2 z + \frac{1}{z}) - 3}$

$∵$ $X \in ℝ, then 2 z + \frac{1}{z} \in ℝ$

$\Rightarrow 2 z + \frac{1}{z} = 2 \bar{z} + \frac{1}{\bar{z}}$ $\Rightarrow 2 (z - \bar{z}) - \frac{z - \bar{z}}{| z |^{2}} = 0$

$∵$ $(z - \bar{z}) (2 - \frac{1}{| z |^{2}}) = 0$

$∵$ $z \neq \bar{z} (given), So, | z |^{2} = \frac{1}{2}$

Solution

Q.
Let $z$ be a complex number with non-zero imaginary part. If $\frac{2 + 3 z + 4 z^{2}}{2 - 3 z + 4 z^{2}}$ is a real number, then the value of $| z |^{2}$ is _______. [2022]

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Solution

Q. Let z be a complex number with non-zero imaginary part. If 2+3z+4z22-3z+4z2 is a real number, then the value of |z|2 is _______. [2022]

Ans. (0.50) Let X=4z2+3z+24z2-3z+2 It can be written as=1+6z4z2-3z+2 ; Now X=1+62(2z+1z)-3 ∵ X∈ℝ, then 2z+1z∈ℝ ⇒2z+1z=2z¯+1z¯⇒2(z-z¯)-z-z¯|z|2=0 ∵ (z-z¯)(2-1|z|2)=0 ∵ z≠z¯ (given), So, |z|2=12

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Q.
Let $z$ be a complex number with non-zero imaginary part. If $\frac{2 + 3 z + 4 z^{2}}{2 - 3 z + 4 z^{2}}$ is a real number, then the value of $| z |^{2}$ is _______. [2022]