Let $z$ be a complex number such that$|z-i|\leq5$, and let $z_1=5+3i$.
Find the minimum and maximum values of $|z_1+iz|$.
The geometric way to do this is easy, just draw a circle of radius $5$ centered at $(0,1)$ and find the minimum and maximum distances from there. But is there a way to do this purely algebraically?
My attempt:
Let $z=a+ib$
$\sqrt{a^2+(b-1)^2}\leq5$
$a^2+b^2-2b+1\leq25\qquad[1]$
Now, $|z_1+iz|=\sqrt{(5-b)^2+(3+a)^2}$
Adding $6a-8b+33$ to $[1]$, we get $|z_1+iz|^2\leq58+6a-8b$
I don't know where to go from here. Please help.