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?
Adding $6a-8b+33$ to $$, we get $|z_1+iz|^2\leq58+6a-8b$
I don't know where to go from here. Please help.