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What happens if we start with an equation of a circle in standard form, (x - h)2 + (y - k)2 = r2 and perform some natural algebra on it?

Expand the squares, collect like terms and simplify.

EXAMPLE 1
Equation to expand	(x - 1)2 + (y - 2)2 = 4
Expand the squares	x2 - 2x + 1 + y2 - 4y + 4 = 4
Collect like terms	x2 + y2 - 2x - 4y + (1 + 4 - 4) = 0
Simplify	x2 + y2 - 2x - 4y + 1 = 0

The equations (x - 1)2 + (y - 2)2 = 4 and x2 + y2 - 2x - 4y + 1 = 0 are equivalent. That is, they both describe a circle centred at (1, 2) and with a radius of 2.

Unfortunatly, not all equations expand as nicely as the one in Example 1 did. Many times we are left with coefficients which are not integers.

If each coefficient happens to be a rational number (a fraction), then we can always find some number A by which to multiply the entire equation and clear all the denominators.

After performing this multiplication by A, we are left with an equation of the form Ax2 + Ay2 + Cx + Dy + F = 0.

EXAMPLE 2
Equation to expand	(x - 1)2 + (y + 1/2)2 = 4
Expand the squares	x2 - 2x + 1 + y2 + y + 1/4 = 4
Collect like terms	x2 + y2 - 2x + y + (1 + 1/4 - 4) = 0
Simplify	x2 + y2 - 2x + y - 11/4 = 0
Multiply everything by 4 to clear the denominator.	4x2 + 4y2 - 8x + 4y - 11 = 0

In Example 2, all 5 equations written on the right hand side are equivalent. That is, the set of points (x, y) which satisfies one of these equations is identical to the set which satisfies any other. That is, each of these equations represents a circle of radius 2 with centre at (1, -1/2).

The equations x2 + y2 - 2x + y - 11/4 = 0 and 4x2 + 4y2 - 8x + 4y - 11 = 0 from Example 2, and the equation x2 + y2 - 2x - 4y + 1 = 0 from Example 1 are each in the form Ax2 + Ay2 + Cx + Dy + F = 0. Any equation in this form is said to be in general form.