Question:
If $\left|Z-\frac{4}{z}\right|=2$, then the maximum value of $\left|Z\right|$ is equal to
If $\left|Z-\frac{4}{z}\right|=2$, then the maximum value of $\left|Z\right|$ is equal to
Updated On: Jul 28, 2022
- $\sqrt{3}+1$
- $\sqrt{5}+1$
- $2$
- $2+\sqrt{2}$
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The Correct Option is B
Solution and Explanation
$\left|Z\right|=\left|\left(Z-\frac{4}{Z}\right)+\frac{4}{Z}\right| \Rightarrow \left|Z\right|=\left|Z-\frac{4}{Z}+\frac{4}{Z}\right|$
$\Rightarrow \left|Z\right|\le \left|Z-\frac{4}{Z}\right|+\frac{4}{\left|Z\right|} \Rightarrow \left|Z\right|\le2+\frac{4}{\left|Z\right|}$
$\Rightarrow \left|Z\right|^{2}-2\left|Z\right|-4\le0$
$\left(\left|Z\right|-\left(\sqrt{5}+1\right)\right)\left(\left|Z\right|-\left(1-\sqrt{5}\right)\right)\le0 \Rightarrow 1-\sqrt{5} \le\left|Z\right|\le\sqrt{5}+1$
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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.
