Question 1206417
.


The problem contains a tip to you to determine the radius.


Since the center is the point  (-2,3),  from it you  IMMEDIATELY conclude that 
the radius of the circle is  2  units,  since the circle touches  y-axis.


As soon as you know the center and the radius,  the rest is just obvious.


The final equation of the circle is


            {{{(x+2)^2}}} + {{{(y-3)^2}}} = {{{2^2}}},


or,  which is the same


            {{{(x+2)^2}}} + {{{(y-3)^2}}} = 4.


===============


Solved.


Never say that the radius is -2, as the other person does.


A radius of a circle  (of any circle)  is always a positive value 
and can not be negative.



\\\\\\\\\\\\\\\\\\\\\



You ask in your post, if your logic works.


No, the logic presented in your post does not work in this problem.


It does not work, because you try to use some info, which is not given in the problem,
and try to make conclusions from false/irrelevant premises.