There is another way to express circle equations apart from its standard form [math]\left(x-h\right)^2+\left(y-k\right)^2=r^2[/math], and that is to express it in the general form [br][center][math]x^2+y^2-2ax-2by=c[/math][/center]where a, b, and c are constants.

In this section, you will learn how to convert the general form of the circle equation to its standard form.[br][i][center][b][size=85]Prerequisite knowledge required: completing the square.[br][color=#38761d](Watch Completing the Square [url=https://www.youtube.com/watch?v=4L3an7mSfm4]Part 1[/url] | [url=https://www.youtube.com/watch?v=sMJ9IXUtffY]Part 2[/url] | [url=https://www.youtube.com/watch?v=UhllPyomgno]Part 3[/url])[/color][/size][/b][/center][/i][br]Suppose we have an equation of a circle in the general form: [math]x^2+y^2-4x+6y=12[/math][br][br][table][br][tr][td][b]Step 1[/b]: Arrange the x's and y's together.[/td][br] [td][right][math]x^2-4x+y^2+6y=12[/math][/right][/td][/tr][br][tr][td][b]Step 2:[/b] Complete the square for variable x.[/td][br] [td][right][math] x^2-4x+4 - 4+y^2+6y =12 [/math][/right][/td][/tr][br][tr][td][/td][td][right][math] (x-2)^2 - 4+y^2+6y =12[/math][/right][/td][/tr][br][tr][td][b]Step 3: [/b]Complete the square for variable y.[/td][br] [td][right][math] (x-2)^2 - 4+y^2+6y + 9 -9 =12 [/math][/right][/td][/tr][br][tr][td][/td][td][right][math] (x-2)^2 - 4+(y+3)^2-9 =12 [/math][/right][/td][/tr][br][tr][td][b]Step 4:[/b] Shift constant terms to the RHS.[/td][br] [td][right][math] (x-2)^2+(y+3)^2 =25 [/math][/right][/td][/tr][br][/table][br]By converting from the general form to the standard form, you can read off quite easily where the coordinates of the centre of the circle is, and also its radius.[br][br]In the example above, we have the center [math](2,-3)[/math] with radius [math]\sqrt{25}=5[/math].[br]