Discuss how to convert the standard form of the equation of a circle to the general form.

Explanation:

The standard form of the equation of a circle is ...

  (x -h)² +(y -k)² = r² . . . . . . circle centered at (h, k) with radius r

The general form of the equation of a circle will be ...

  Ax² +Bxy +Cy² +Dx +Ey +F = 0 . . . . . A = C for a circle

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The former is converted to the latter by eliminating parentheses, subtracting r² and collecting terms.

  x² +y² -2hx -2ky +(h² +k² -r²) = 0

This shows you that ...

  A = C = 1

  B = 0

  D = -2h

  E = -2k

  F = h² +k² -r²

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In general form, we like to have the coefficients be mutually prime integers and the leading coefficient be positive. That is not always possible, depending on the circle being represented.

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