Question

If$|z-i|\le 2$ and $z_0=5+3i$ then the maximum value of $|iz+z_0|$ is

• A. $2+\sqrt{31}$
• B. $7$
• C. $\sqrt{31}-2$
• D. None of these

Answer 1

The correct option is B $7$

$|z-i|\le 2$
z lies on and in the inner part of the circle $|z-i|\le 2$
$z_0=5+3i$
From the fig:
$max\left(|iz+z_0|\right)$ will be the distance between point A & point B=[radius of circle (r)] + [distance between point A and center of the circle (d(AC))].
$max\left(|iz+z_0|\right)=max\left(|z-i{z}_0|\right)=max\left(|z-(-3+5i)|\right)=r+d\left(AC\right)$
$\Rightarrow max\left(|iz+z_0|\right)=2+\sqrt{{(0-(-3))}^2+{(1-5)}^2}=7$
Ans: B