Question 967614
To write the equation of any circle, you need a center and a radius. So, if (a,b) is the center and r is the radius of the circle, the circle equation would be 
                         {{{(x-a)^2 + (y-b)^2 = r^2}}}
In this problem, you have 2 pieces of information. One, 2 points one the circle and two, a tangent. Let the coordinates of the center be O(a,b) and the radius be r. 
Since point A and point B are points on the circle, distance between O and A will be equal to O and B which in turn would be equal to r.
                         OA = OB = r
                       => {{{sqrt((a-1)^2+(b-2)^2) = r}}}....... (1)
                          {{{sqrt((a-1)^2+(b-(-2))^2) = r}}}.... (2)
Also, since x+2y+5 = 0 touches the circle, it is a tangent to the circle. and hence, distance between centre O and the line will also be equal to r.
                => {{{sqrt((a+2b+5)/sqrt(1^2+2^2)) = r}}}......... (3)

For the 3 variables a,b,r, you have 3 equations. Solve them and you will get the answer.

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