Question

If $z$ is a complex number satisfying $|z-3-2i|\le 1$, then the maximum value of $|iz+2|$ is

• A. $16$
• B. $24$
• C. $4$
• D. $32$

Answer 1

The correct option is C $4$

Assuming $z=x+iy$
$|z-3-2i|\le 1\Rightarrow (x-3{)}^2+(y-2{)}^2\le 1$


Now finding the maximum,
$|iz+2|=|i(x+iy)+2|=|ix-y+2|=\sqrt{x^2+(y-2{)}^2}=\sqrt{(x-3{)}^2+(y-2{)}^2+6x-9}\le \sqrt{1+6x-9}=\sqrt{6x-8}$
Now maximum value will occur when $x=4$
$\sqrt{(6\times 4)-8}=4$