A complex number $z$ is the said to be unimodular if $|z|=1 .$ Suppose $z_{1}$ and $z_{2}$ are
complex number such that $\frac{z_{1}-2 z_{2}}{2-z_{1} z_{2}}$ is unimodular and $z _{2}$ is not unimodular. Then the point $z_{1}$ lies on a :
- Straight line parallel to x-axis
- Straight line parallel to y-axis
- Circle of radius 2
- Circle of radius $\sqrt{2}$
The Correct Option is C
Solution and Explanation
$\left(z_{1}-2 z_{2}\right)\left(\bar{z}_{1}-2 \bar{z}_{2}\right)=\left(2-z_{1} \bar{z}_{2}\right)\left(2-\bar{z}_{1} z_{2}\right)$
$\left|z_{1}\right|^{2}-2 z_{1} \bar{z}_{2}-2 z_{2} \bar{z}_{1}+4\left|z_{2}\right|^{2} $
$=4-2 \bar{z}_{1} z_{2}-2 z_{1} \bar{z}_{2} +\left|z_{1}\right|^{2}\left|z_{2}\right|^{2}$
$\left|z_{1}\right|^{2}\left|z_{2}\right|^{2}-\left|z_{1}\right|^{2}-4\left|z_{2}\right|^{2}+4=0$
$\left(\left|z_{1}\right|^{2}-4\right)\left(\left|z_{2}\right|^{2}-1\right)=0$
$\Rightarrow\left|z_{1}\right|=2$
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Concepts Used:
Complex Numbers and Quadratic Equations
Complex Number: Any number that is formed as a+ib is called a complex number. For example: 9+3i,7+8i are complex numbers. Here i = -1. With this we can say that i² = 1. So, for every equation which does not have a real solution we can use i = -1.
Quadratic equation: A polynomial that has two roots or is of the degree 2 is called a quadratic equation. The general form of a quadratic equation is y=ax²+bx+c. Here a≠0, b and c are the real numbers.
