LEARNING OBJECTIVES
What you will be able to do
- identify a circle's centre and radius from any standard form
- solve line–circle intersections and classify them
- find equations of tangents without calculus
- analyse two circles and their common chord
AT A GLANCE
INTRODUCTION · THE BIG IDEA
Translate geometric facts about centres, radii, chords and tangents into equations that can be solved systematically.
A circle equation records one distance condition: every point is the same distance from its centre. Completing the square uncovers this geometry when the equation is expanded.
Intersections reduce to simultaneous equations. Discriminants classify line contact, while subtraction is especially efficient for two-circle problems.
SECTION 01
Circle equations
In general form x² + y² + 2gx + 2fy + c = 0, the centre is (−g, −f). Complete both squares to obtain the radius and confirm that r² is positive.
Recover centre and radius
- Start with x² + y² − 6x + 4y − 12 = 0.
- Group and complete squares: (x − 3)² − 9 + (y + 2)² − 4 − 12 = 0.
- Rearrange: (x − 3)² + (y + 2)² = 25.
- Read the centre and radius.
Answer: Centre (3, −2), radius 5.
SECTION 02
A line meeting a circle
Substitute the line equation into the circle to form a quadratic in one coordinate. Solve for intersection points or use the discriminant to classify the contact.
| Discriminant | Geometry |
|---|---|
| Δ > 0 | two points: the line is a chord |
| Δ = 0 | one repeated point: the line is tangent |
| Δ < 0 | no real intersection |
SECTION 03
Tangents without calculus
The radius to a point of contact is perpendicular to the tangent. Find the centre-to-point gradient, take its negative reciprocal, then use the contact point in point-gradient form.
DETAILED EXPLANATION
- A vertical radius produces a horizontal tangent.
- If the contact point is unknown, combine the tangent condition with the circle.
- Verify that a proposed point lies on the circle first.
SECTION 04
Two circles and the common chord
Subtracting the two equations cancels x² and y², leaving the radical axis or common-chord line. Solve that line with either circle to find intersection points.
DETAILED EXPLANATION
- Two intersections occur when |r₁ − r₂| < d < r₁ + r₂.
- No intersection also occurs when one circle lies inside the other without touching.
- Concentric unequal circles have no common chord.
- Distance tests can classify circles without solving coordinates.
QUICK CHAPTER SUMMARY
The ideas to carry forward
- Standard form shows centre and radius immediately.
- Complete squares to convert general form.
- A line–circle substitution produces a quadratic.
- A tangent is perpendicular to the radius at contact.
- Subtracting circle equations gives their common-chord line.
QUICK REVISION CHECKLIST
Can you do each of these without your notes?
- identify a circle's centre and radius from any standard form
- solve line–circle intersections and classify them
- find equations of tangents without calculus
- analyse two circles and their common chord