An expression that can be represented in the form (x - a)^2 + (y -b)^2 = r^2 is the equation of a circle with center at (a, b) and radius r. It has to be of the form given and not just one that has x^2 and y^2 terms as even the equation of an ellipse has x^2 and y^2.
The given equation x^2 − 4x + y^2 − 2y = 11
=> x^2 - 4x + 4 + y^2 - 2y + 1 = 11 + 4 + 1
=> (x - 2)^2 +( y - 1)^ = 4^2
This is a circle with center (2,1) and radius 4
To prove that the given equation is the equation of a circle, we must recognize this form,
(x − a)^2 + (y − b)^2 = r^2.
a and b are the coordinates of the center of circle and r is the radius.
Therefore, we'll create perfect squares in both x and y.
To complete the square in x, we'll add 4 both sides.
To complete the square in y, we'll add 1 both sides.(x^2 − 4x + 4) + (y^2 − 2y + 1) = 11 + 4 + 1 (x − 2)^2 + (y − 1)^2 = 16.
This is the equation of a circle of radius 4, whose center is at (2, 1).