Position of two circles — A/L Combined Mathematics

What must I remember about Position of two circles?

• Relative positions of two non-concentric circles: Let C1, C2 be centers, r1, r2 be radii (assume r1 ≥ r2), and d = C1C2 be the distance between centers.
• Disjoint / External (4 common tangents): d > r1 + r2 (two direct external tangents, two transverse internal tangents).
• Touching Externally (3 common tangents): d = r1 + r2 (point of contact divides C1C2 internally in ratio r1 : r2).
• Intersecting at two real points (2 common tangents): |r1 - r2| < d < r1 + r2 (two direct external tangents only).
• Touching Internally (1 common tangent): d = r1 - r2 (point of contact divides C1C2 externally in ratio r1 : r2); One inside other / Disjoint internally: d < r1 - r2 (0 common tangents); Concentric circles: d = 0.

Test yourself (1 minute)

1. Intersection of Direct Common Tangents

   Meet at external center of similitude P dividing C1C2 externally in ratio r1 : r2: P = (r1*C2 - r2*C1) / (r1 - r2).

2. Intersection of Transverse Common Tangents

   Meet at internal center of similitude Q dividing C1C2 internally in ratio r1 : r2: Q = (r1*C2 + r2*C1) / (r1 + r2).

3. Lengths of common tangents

   Direct common tangent length = √[d² - (r1 - r2)²]; Transverse common tangent length = √[d² - (r1 + r2)²].

Practise Position of two circles questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Pure Mathematics, Unit 19

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Year 1 · Term 3