Question 1041650
The line that passes through the point (-2,-2) and the center (1,2) has equation 

{{{y-2 = ((2--2)/(1--2))(x-1)}}}  <==> {{{y = (4/3)x + 2/3}}},

after simplifying.

This line will intersect the circle in two (diametrically) opposite points on the circle, one of which is the farthest from the point (-2,-2), and the other the nearest to the point (-2,-2).

To find out the points, substitute {{{y = (4/3)x + 2/3}}} into 
{{{(x-1)^2 + (y-2)^2 = 4}}}.

===>{{{(x-1)^2 + ((4/3)x + 2/3-2)^2 = (x-1)^2 +  ((4/3)x-4/3)^2 = 4}}}

==> {{{(x-1)^2 +  (16/9)(x-1)^2 = 4}}}

==> {{{(25/9)(x-1)^2 = 4}}}  ==> {{{(x-1)^2 = 36/25}}}

==> x = 11/5 or -1/5.

Incidentally, these are the x-coordinates of the two diametrically opposite points on the circle that are farthest and nearest the point (-2,-2), respectively.  The corresponding y-value is y = 2/5.  The distance of this point (-1/5,2/5) from (-2,-2) is {{{sqrt((-1/5--2)^2+(2/5--2)^2) = highlight(3)}}}.