we know that
the standard form of the equation of the circle is
\((x-h) ^{2} +(y-k) ^{2} =r ^{2}\)
where
(h,k) is the center
r is the radius
case a)
\(x^{2} + y^{2} -8x-8y+23=0 \\ (x-4) ^{2} +(y-4)^{2} = 3^{2}\)
Is a circle
with radius r=\(3\) units
and center \((4,4)\)
case b)
\(x^{2} + y^{2} -8x-8y+32=0 \\ (x-4) ^{2} +(y-4) ^{2} =0\)
Is not a circle
case c)
\(x^{2} + y^{2} -4x-y+23=0 \\ (x+2) ^{2} +(y-2) ^{2} =-15\)
Is not a circle
case d)
\(x^{2} + y^{2} +4x+4y+9=0 \\ (x+2) ^{2} +(y+2) ^{2} =-1\)
Is not a circle
therefore
the answer is
\(x^{2} + y^{2} -8x-8y+23=0\)
