Question 1122329
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The equation  |z+1+2i| = 3  represents the set of points in the complex plane that are remoted in 3 units from the point (-1,-2).


It is the circle of the radius 3 with the center at the point (-1,-2).


So the problem asks to find the minimum and the maximum distance from the point (3,-1) to this circle.


The distance from the center of the circle (-1,-2) to the point (3,-1)  is


    {{{sqrt((3-(-1))^2 + ((-1) - (-2))^2)}}} = {{{sqrt(4^2+1^2)}}} = {{{sqrt(17)}}}.


This distance is greater than 3, so the point (3,-1) lies outside that circle.


Now, it is very simple to find the maximum distance and the minimum distance from the given point to the circle.

Simply connect the center  (-1,-2) with the point (3,-1) by the straight line.


The minimum will be  {{{sqrt(17)}}}-3;  the maximum will be {{{sqrt(17)}}} + 3.
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