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  4. Let z1, z2 be two complex numbers represented by points on the circle |z1|=1 and |z2|=2 respectively, then -

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Q. Let $z_1, z_2$ be two complex numbers represented by points on the circle $\left|z_1\right|=1$ and $\left|z_2\right|=2$ respectively, then -

Complex Numbers and Quadratic Equations

Solution:

Since $z_1$ and $z_2$ lie on $|z|=1$ and $|z|=2$
then $\left|z_1\right|=1$ and $\left|z_2\right|=2$
$\left|2 z_1+z_2\right| \leq 2\left|z_1\right|+\left|z_2\right| \leq 4$
$\max \left|2 z_1+z_2\right|=4$
$\left|z_1-z_2\right| \geq|| z_1|-| z_2||=|1-2|=1$
$\min \left|z_1-z_2\right|=1$
$\left|z_2+\frac{1}{z_1}\right| \leq\left|z_2\right|+\frac{1}{\left|z_1\right|}=2+1=3$
$\left|z_2+\frac{1}{z_1}\right| \leq 3$