The equation of a circle is given by \(x^{2} + y^{2} - 4x - 2y - 3\). Find the radius and the coordinates of its centre.

Answer Details

The equation of a circle in standard form is \((x-a)^2 + (y-b)^2 = r^2\), where \((a,b)\) are the coordinates of the center and \(r\) is the radius. We need to rewrite the given equation in this standard form by completing the square for both the \(x\) and \(y\) terms. \begin{align*} x^{2} + y^{2} - 4x - 2y - 3 &= 0\\ (x^{2} - 4x) + (y^{2} - 2y) &= 3\\ (x^{2} - 4x + 4) + (y^{2} - 2y + 1) &= 3 + 4 + 1\\ (x-2)^{2} + (y-1)^{2} &= 8 \end{align*} Comparing with the standard form, we see that the center of the circle is \((2,1)\) and the radius is \(\sqrt{8} = 2\sqrt{2}\). Therefore, the answer is: the radius of the circle is \(2\sqrt{2}\) and the coordinates of its center are \((2,1)\).