show that is a circleshow that something is the equation of a circle x² − 4x + y² − 2y = 11

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justaguide | College Teacher | (Level 2) Distinguished Educator

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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

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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).

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