Question 792682
Mostly a conic-section concept and symbolism exercise.  First, you want to Complete-the-Square for the x and the y, and convert the equation to standard form.  You will obtain something a little complicated, but you can use the formula.


{{{x^2 + y^2 + 2gx + 2fy +c = 0}}} as given.
{{{x^2+2gx+y^2+2fy+c=0}}}
The missing square term for x and for y are, in that order, {{{(2g/2)^2=g^2}}} and {{{(2f/2)^2=f^2}}}.  This will allow you to factor.
{{{(x^2+2gx+g^2)+(y^2+2fy+f^2)+c-g^2-f^2=0}}}
{{{highlight((x+g)^2+(y+f)^2=g^2+f^2-c)}}}


According to that standard form equation for a circle, the center is at (-g,-f) and the radius is {{{sqrt(g^2+f^2-c)}}}.


You should now be able to work through to the necessary conclusions.