Position of two circles — A/L Combined Mathematics

The relative geometric position of two circles with centres C₁, C₂ and radii r₁, r₂ is governed by comparing the distance between their centres d = C₁C₂ against the sum (r₁ + r₂) and difference |r₁ - r₂| of their radii. Five distinct cases arise: (1) Completely outside each other: d > r₁ + r₂ (4 common tangents: 2 direct, 2 transverse); (2) Touching externally: d = r₁ + r₂ (3 common tangents); (3) Intersecting at two real points: |r₁ - r₂| < d < r₁ + r₂ (2 direct common tangents); (4) Touching internally: d = |r₁ - r₂| (1 common tangent); (5) One inside the other without touching: d < |r₁ - r₂| (no common tangents).

Check before you answer (A/L Pure Maths Paper 1)

1. Centres and Radii Extraction: Accurately extracting C₁, C₂, r₁, and r₂ from the general circle forms before calculating distance d.
2. Point of Contact Coordinates: Using internal division formula (r₁ : r₂) for external touch and external division for internal touch to locate contact points.
3. Common Tangent Formulations: Finding intersection points of common tangents along the line of centres to determine direct and transverse tangent equations.