Find the centre and radius of the circle with equation: x^2 + y^2 -4x +8y = 5, and determine whether the point (7,-4) lies on the circle.

First we complete the square on the equation of the circle to obtain: (x-2)^2 -4 + (y+4)^2 -16 = 5. Re arrange : (x-2)^2  + (y+4)^2 = 25 General equation of a circle: (x-a)^2 + (y-b)^2 = r^2, with centre (a,b) and radius r. Therefore centre = (2,-4) and radius = sqrt(25) = 5. The point (7,-4) does lie on circle as (7-2)^2 + (-4+4)^2 = 25