If (a,b) is the centre and c is the radius of the circle x^{2}+y^{2}+2x+6y+1=0, then what is the value of a^{2}+b^{2}+c^{2}?

Explanation

Comparing x^2+y^2+2x+6y+1=0 with the standard form x^2+y^2+2gx+2fy+k=0, we have 2g=2 \implies g=1 and 2f=6 \implies f=3. The centre (a,b) = (-g,-f) = (-1,-3). The radius c = \sqrt{g^2+f^2-k} = \sqrt{1^2+3^2-1} = 3. Therefore, a^2+b^2+c^2 = (-1)^2 + (-3)^2 + 3^2 = 1 + 9 + 9 = 19.

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