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Let $\mathrm{z}_{1}$ and $\mathrm{z}_{2}$ be two non-zero complex numbers such that
$\left|z_{1}\right|=\left|z_{2}\right|=\left|\frac{1}{z_{1}}+\frac{1}{z_{2}}\right|=2$
What is the value of $\left|z_{1}+z_{2}\right|$ ?
Options:
$\left|z_{1}\right|=\left|z_{2}\right|=\left|\frac{1}{z_{1}}+\frac{1}{z_{2}}\right|=2$
What is the value of $\left|z_{1}+z_{2}\right|$ ?
Solution:
2835 Upvotes
Verified Answer
The correct answer is:
8
As given $\left|z_{1}\right|=\left|z_{2}\right|=\left|\frac{1}{z_{1}}+\frac{1}{z_{2}}\right|=2$
$\left|\frac{1}{z_{1}}+\frac{1}{z_{2}}\right|=2 \Rightarrow \frac{z_{2}+z_{1}}{z_{1} z_{2}}=2$
$\Rightarrow \frac{\left|z_{1}+z_{2}\right|}{\left|z_{1}\right|\left|z_{2}\right|}=2$
$\Rightarrow\left|z_{1}+z_{2}\right|=2\left|z_{1} \| z_{2}\right|$
$\Rightarrow\left|z_{1}+z_{2}\right|=2.2 .2$
$\Rightarrow\left|z_{1}+z_{2}\right|=8$
$\left|\frac{1}{z_{1}}+\frac{1}{z_{2}}\right|=2 \Rightarrow \frac{z_{2}+z_{1}}{z_{1} z_{2}}=2$
$\Rightarrow \frac{\left|z_{1}+z_{2}\right|}{\left|z_{1}\right|\left|z_{2}\right|}=2$
$\Rightarrow\left|z_{1}+z_{2}\right|=2\left|z_{1} \| z_{2}\right|$
$\Rightarrow\left|z_{1}+z_{2}\right|=2.2 .2$
$\Rightarrow\left|z_{1}+z_{2}\right|=8$
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