The equation of a circle is \(3x^{2} + 3y^{2} + 24x - 12y = 15\).

Explanation

The equation of a circle is given as: \((x - a)^{2} + (y - b)^{2} = r^{2}\)

Expanding, we have: \(x^{2} + y^{2} - 2ax - 2by + a^{2} + b^{2} = r^{2}\)

\(\implies x^{2} + y^{2} - 2ax - 2by = r^{2} - a^{2} - b^{2}\)

Comparing with the given equation: \(3x^{2} + 3y^{2} + 24x - 12y = 15\)

Making the coefficients of \(x^{2}\) and \(y^{2}\) = 1, we have

\(x^{2} + y^{2} + 8x - 4y = 5\)

\(2a = -8 \implies a = -4\)

\(2b = 4 \implies b = 2\)

\(r^{2} - a^{2} - b^{2} = 5 \implies r^{2} = 5 + (-4)^{2} + (2)^{2} = 5 + 16 + 4 = 25\)

\(\therefore r = 5\)