Circles and tangents
Complete the square to read off a circle’s centre and radius.
In (x − a)² + (y − b)² = r², the centre is (a,b) and r is nonnegative. If the right side becomes negative after completing squares, there is no real circle.
The radius to a point of tangency is perpendicular to the tangent. Find its gradient and take the negative reciprocal, remembering the vertical and horizontal cases. Check that the proposed point actually lies on the circle.
Worked example
Find the tangent to x² + y² = 25 at (3,4).
- The radius gradient is 4/3.
- The tangent gradient is −3/4.
- Use y − 4 = −3(x − 3)/4.
Answer: 3x + 4y = 25
Revise first: Straight lines and gradients.
Practise circles and tangents
Course mapping
These specification references show where the topic occurs. The questions cover only some parts of each topic.
- 8365 · core · 3.8: General circle equations
- 7M20 · core · A2.10: General circle equations
- 6993 · core · CG2: General circle equations
- 8365 · core · 3.8: Centre and radius from a circle equation
- 7M20 · core · A2.10: Centre and radius from a circle equation
- 6993 · core · CG2: Centre and radius from a circle equation
- 7M20 · core · A2.10: Centre and radius from a circle equation
- 8365 · core · 3.9: A tangent at a specified circle point
- 7M20 · core · A2.10: A tangent at a specified circle point
- 7M20 · core · A2.10: A tangent at a specified circle point
- 8365 · core · 3.7: Chord length from its distance to the centre
Next practice: Tangents and normals.