What is the minimum distance between centers of two circles having radii 5 cm and 3 cm such that exactly three common tangents exist?

Correct — A.
Exactly three common tangents exist when the circles touch externally: two direct tangents plus one at the point of contact.

Touching externally: distance between centres = r₁ + r₂
= 5 + 3 = 8 cm → option (a).

Why the others are wrong

• (b)
  2 cm — 2 cm = 5 − 3 makes the circles touch internally, which leaves a single common tangent, at the point of contact.
• (c)
  5 cm — 5 cm lies between 5 − 3 = 2 and 5 + 3 = 8, so the circles cut at two points and share just two common tangents.
• (d)
  10 cm — 10 cm is more than 5 + 3 = 8, so the circles are separate and have four common tangents: two direct, two transverse.

The number of common tangents depends on the distance d between the centres, measured against r₁ + r₂ = 8 and r₁ − r₂ = 2.

As d shrinks, the count runs 4, 3, 2, 1, 0: separate, touching externally, intersecting, touching internally, one circle inside the other. Three occurs at a single distance, d = 8.

Key facts

• d > r₁ + r₂ (separate circles): 4 common tangents.
• d = r₁ + r₂ (touching externally): 3 common tangents.
• r₁ − r₂ < d < r₁ + r₂ (intersecting): 2 common tangents.
• d = r₁ − r₂ (touching internally): 1 common tangent.

Common traps

• Using r₁ − r₂ = 2 cm, the internal-touching distance. That gives one common tangent, not three.
• Swapping the two touching cases: touching externally gives 3 tangents, touching internally gives 1.

No directly related past PYQ was found.