Equations of circles
- A circle with centre (0, 0) and radius r has equation x² + y² = r².
- A circle with centre (a, b) and radius r has equation (x − a)² + (y − b)² = r². For (x − 2)² + (y + 5)² = 49, the centre is (2, −5) and the radius is 7.
- Watch the signs: (y + 5) means b = −5.
Completing the square
- If the equation is expanded, complete the square for x and for y.
- x² + y² − 6x + 4y − 12 = 0 becomes (x − 3)² − 9 + (y + 2)² − 4 − 12 = 0, so (x − 3)² + (y + 2)² = 25. The centre is (3, −2) and the radius is 5.
Tangents and circle facts
- A tangent is perpendicular to the radius at the point where it touches the circle.
- Tangent to x² + y² = 25 at (3, 4): the radius has gradient 4/3, so the tangent has gradient −3/4. y − 4 = −(3/4)(x − 3), which gives 3x + 4y = 25.
- The angle in a semicircle is 90°, and the perpendicular from the centre to a chord bisects the chord.
- To test a point, substitute it: for x² + y² = 36, the point (5, 5) gives 50 > 36, so it is outside the circle.
Key terms
- Radius
- The distance from the centre of a circle to its edge.
- Centre
- The point equally distant from every point on the circle.
- Tangent
- A straight line that touches a circle at one point.
- Chord
- A straight line joining two points on a circle.
- Semicircle
- Half a circle.