Question

The greatest and least value of $\left|z\right|$ if $z$ satisfies $|z-5+5i|$ $\le 5$ are

• A. $10$ , $5\sqrt{2}$
• B. $5\sqrt{2}$ , $5$
• C. $10$ , $0$
• D. $5+5\sqrt{2}$ , $5\sqrt{2}-5$

Answer 1

The correct option is D $5+5\sqrt{2}$ , $5\sqrt{2}-5$

$|z-5+5i|$ $\le 5$ represents equation of circle, where centre of circle is (5,-5)

Maximum value of $z=c+\sqrt{a^2+b^2}$

Maximum value of $z=c-\sqrt{a^2-b^2}$

Here, $a=5,b=-5$ and $c=5$

Maximum value of $|z|=|5+\sqrt{25+25}|$

$=|5+\sqrt{50}|$

$=5+5\sqrt{2}$

Minimum value of $|z|=|5-\sqrt{50}|$

$=|5-5\sqrt{2}|$

$=5\sqrt{2}-5$