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 ___ .
Approach Solution - 1
Approach Solution -2
\(1+ \frac{6}{\frac{2}{z}-3+4z}\epsilon R\)
\(2z+\frac{1}{z}\epsilon R\)
\(2z+\frac{1}{z}=2\bar{z}+\frac{1}{\bar{z}}\)
\((2\bar{z}z-1)(z-\bar{z})=0\)
\(|z|^2=\frac{1}{2}=0.50\)
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Concepts Used:
Complex Number
A Complex Number is written in the form
a + ib
where,
- “a” is a real number
- “b” is an imaginary number
The Complex Number consists of a symbol “i” which satisfies the condition i^2 = −1. Complex Numbers are mentioned as the extension of one-dimensional number lines. In a complex plane, a Complex Number indicated as a + bi is usually represented in the form of the point (a, b). We have to pay attention that a Complex Number with absolutely no real part, such as – i, -5i, etc, is called purely imaginary. Also, a Complex Number with perfectly no imaginary part is known as a real number.