Circles centred at the origin
- The equation of a circle with centre (0, 0) and radius r is x² + y² = r².
- x² + y² = 25 has radius 5. x² + y² = 20 has radius √20 = 2√5.
- A point is on the circle if its coordinates fit the equation: (3, 4) is on x² + y² = 25 because 9 + 16 = 25.
Tangents
- A tangent touches the circle at one point, and it is perpendicular to the radius at that point.
- To find a tangent at P: find the gradient of the radius from the origin to P, find the perpendicular gradient (the negative reciprocal), then use y = mx + c through P.
- At (2, 4) on x² + y² = 20: the radius has gradient 4 ÷ 2 = 2, so the tangent has gradient −1/2. 4 = −½ × 2 + c gives c = 5, so the tangent is y = −½x + 5.
Lines meeting circles
- Substitute the equation of the line into the equation of the circle and solve the quadratic.
- Two solutions: the line crosses the circle twice. One solution: the line is a tangent. No solutions: the line misses the circle.
Key terms
- Equation of a circle
- x² + y² = r², for a circle with centre (0, 0) and radius r.
- Radius
- The distance from the centre of a circle to its edge.
- Tangent
- A straight line that touches a circle at exactly one point.
- Perpendicular
- At right angles to.
- Negative reciprocal
- The gradient of a perpendicular line: flip the fraction and change the sign.
- Origin
- The point (0, 0).