Question

If $|Z-i|\le 2$ and $Z_1=5+3i$, then the maximum value of $|iZ+Z_1|$ is

• A. $7+\sqrt{13}$
• B. $7+\sqrt{12}$
• C. $7$
• D. $9$

Answer 1

The correct option is C $7$

We have,
$|Z-i|\le 2$
Let $z=x+iy$, then
$\Rightarrow x^2+(y-1{)}^2\le 4$
So, $Z$ lies on or inside the circle, whose center is $(0,1)$ and radius is $2\text{units}$

Now,
$|iZ+Z_1|=|iZ-i^2{Z}_1|=|i(Z-i(5+3i\left)\right)|=|i||Z-(-3+5i)|=|Z-(-3+5i)|(\because |i|=1)$

Point $(-3,5)$ lies outside the circle, so the maximum value of $|iZ+Z_1|$
$=\sqrt{(0+3{)}^2+(1-5{)}^2}+2=7$