Question

Assertion :The greatest value of the moduli of complex numbers $z$ satisfying the equation $|z-\frac{4}{z}|=2$ is $\sqrt{5}+1$ Reason: For any two complex number $z_1$ and $z_2,$ $|z_1-z_2|\ge \left|z_1\right|-\left|z_2\right|$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

We have, $2=|z-\frac{4}{z}|\ge \left|z\right|-\left|\frac{4}{z}\right|$

$\Rightarrow \left|z\right|-\left|\frac{4}{z}\right|\le 2\Rightarrow {\left|z\right|}^2-2\left|z\right|-4\le 0\Rightarrow {(\left|z\right|-1)}^2-5\le 0\Rightarrow {(\left|z\right|-1)}^2\le 5\Rightarrow \left|z\right|-1\le \sqrt{5}\Rightarrow \left|z\right|\le \sqrt{5}+1$

Hence, the greatest value of $\left|z\right|$ is $\sqrt{5}+1$

Hence, assertion and reason both are correct. And reason is correct explanation for assertion.