A circle is the set of all points at a fixed distance \(r\) from a fixed centre \(C(a,b)\). Applying the distance formula to a moving point \(P(x,y)\) on the circle gives \((x-a)^2+(y-b)^2=r^2\) — the same Pythagoras idea you used for the distance between two points.
The two forms of a circle's equation
| Form | Equation | Example |
|---|---|---|
| Standard form: centre \((a,b)\) and radius \(r>0\) can be read off directly. | \((x-a)^2+(y-b)^2=r^2\) | \((x-4)^2+(y+2)^2=9\) Centre \((4,-2)\) Radius \(=3\) |
| General form: no brackets; complete the square to find the centre and radius. | \(x^2+y^2+Dx+Ey+F=0\) | \(x^2+y^2-10x+6y+18=0\) |
From general form to standard form: complete the square
| Step | Working |
|---|---|
| Group the \(x\)-terms and the \(y\)-terms. | \(x^2+y^2-10x+6y+18=0\) \((x^2-10x)+(y^2+6y)+18=0\) |
| Complete each square. | \((x-5)^2-25+(y+3)^2-9+18=0\) |
| Collect the constants on the right. | \((x-5)^2+(y+3)^2=16\) |
| Read off the centre and the radius. | Centre \((5,-3)\) Radius \(=\sqrt{16}=4\) |
To go from standard form back to general form, expand the brackets and collect every term on one side.
The centre coordinates have the opposite signs to the numbers inside the brackets: \((x-5)^2+(y+3)^2=16\) has centre \((5,-3)\), not \((-5,3)\). After completing the square the right-hand side is \(r^2\), so it must be positive for a real circle, and the radius is its positive square root.
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