A circle has the equation x^2 + y^2 - 4x + 10y - 115 = 0. Express the equation in the form (x - a)^2 + (y - b)^2 = k, and find the centre and radius of the circle.

Rearrange the equation so that the x terms and y terms bracketed together, like so: (x^2 - 4x) + (y^2 + 10y) - 115 = 0. Complete the square on both sets of brackets, giving (x - 2)^2 - 4 + (y + 5)^2 - 25 - 115 = 0. Move the constants to the right hand side, giving (x - 2)^2 + (y - (-5))^2 = 144. This is the equation in the desired form. The centre of a circle is given by the coordinates (a,b), and so the centre of this circle is (2,-5). The radius of the circle is the root of the constant k. The equation of a circle often uses r^2 in place of k. This is a common trick that catches out many students. In this case, k = 144, and so the radius r = root(144) = 12.