Question 1122329
Grab a pencil and blank paper if you can or do it in your imagination.
We now have a complex number(z+1+2i), it's magnitude is 3. So we draw a circle with radius 3 and center on (0,0), on the complex plane. This represents the endpoints of all possible complex number(z+1+2i)s.


To get the endpoint of (z-3+i), we must add (z+1+2i) with (-4-i), which means we must move its endpoint 4 units in the negative imaginary axis' direction and 1 in the nagative real axis' direction. Try do it with random endpoints in the circle-which one gives the longest magnitude(distance to the origin) and smallest?


You'll see the answer is pretty simple. The maximum and minimum magnitude both occur when the endpoints of (z+1+2i) are on the line which (-4-i) is on. Now time for calculations.


Maximum {{{sqrt(17)+3}}}
Minimum {{{sqrt(17)-3}}}