Two forms
A standard method from Equations of circles: the steps and worked examples.
Steps
| Form | Equation | Centre | Radius |
|---|---|---|---|
| standard | (x − h)² + (y − k)² = r² | (h, k) | r |
| general | x² + y² + Dx + Ey + F = 0 | (−D/2, −E/2) | √[(D/2)² + (E/2)² − F] |
Why: the standard form is the distance formula: every point (x, y) is r away from (h, k).
- Standard form: read the centre and radius directly. Watch the signs: (x − 3)² + (y + 2)² = 25 has centre (3, −2).
- General form: make the x² and y² coefficients 1 first, then centre (−D/2, −E/2) and the radius formula.
- To change standard to general, expand and collect terms. To change general to standard, complete the square.
Worked examples
Example 1: Two forms: centre (3, −2), radius 5. Write the equation in standard form and in general form.
(x − 3)² + (y + 2)² = 25
x² − 6x + 9 + y² + 4y + 4 = 25
x² + y² − 6x + 4y − 12 = 0For the whole topic, read Equations of circles: key points, common mistakes and how the DSE tests it.