Centre from General Form

Centre of x² + y² + 2gx + 2fy + c = 0. Read off as the negatives of half the linear coefficients.

Starting from $x^2 + y^2 + 2gx + 2fy + c = 0$, complete the square in $x$ and $y$:

$$(x^2 + 2gx + g^2) + (y^2 + 2fy + f^2) = g^2 + f^2 - c$$ $$(x + g)^2 + (y + f)^2 = g^2 + f^2 - c$$

Comparing with $(x - h)^2 + (y - k)^2 = r^2$:

$$h = -g, \quad k = -f$$

Therefore the centre is $(-g, -f)$.

The centre coordinates are the negatives of half the respective linear coefficients in the original equation.