Question

If $z_1=7+24i$ and $|z_2|=5,$ then

• A. Maximum value of $|z_1+z_2|$ is $30$
• B. Maximum value of $|z_1+z_2|$
  $=$ Maximum value of $|z_1-z_2|$
• C. Minimum value of $\left|\frac{z_1}{z_2+\frac{5}{z_2}}\right|$ is $\frac{25}{6}$
• D. Maximum value of $\left|\frac{z_1}{z_2+\frac{5}{z_2}}\right|$ is $5$

Answer 1

Answer: (A), (B), (C)

The correct options are
A Maximum value of $|z_1+z_2|$ is $30$
B Maximum value of $|z_1+z_2|$
$=$ Maximum value of $|z_1-z_2|$
C Minimum value of $\left|\frac{z_1}{z_2+\frac{5}{z_2}}\right|$ is $\frac{25}{6}$


$max|z_1+z_2|=max|z_1-(-z_2)|=OA+r=30$
$max|z_1-z_2|=OA+r=30$
$\Rightarrow max|z_1+z_2|=max|z_1-z_2|$

$\left|\frac{z_1}{z_2+\frac{5}{z_2}}\right|=\frac{|z_1|}{|z_2+\frac{5}{z_2}|}$

Let $z_2=5e^{i\theta }$

$\frac{|z_1|}{|z_2+\frac{5}{z_2}|}$

$=\frac{25}{|5e^{i\theta }+e^{-i\theta }|}=\frac{25}{|6\cos \theta +4i\sin \theta |}=\frac{25}{\sqrt{36{\cos }^2\theta +16{\sin }^2\theta }}=\frac{25}{\sqrt{20{\cos }^2\theta +16}}$

$\therefore min\left|\frac{z_1}{z_2+\frac{5}{z_2}}\right|=\frac{25}{6}$
and $max\left|\frac{z_1}{z_2+\frac{5}{z_2}}\right|=\frac{25}{4}$