Three non-overlapping, non-touching circles are drawn in a plane. What is the maximum number of distinct common tangents that can be drawn to them in total?

Correct — D. Common tangents are drawn to two circles at a time, so count the pairs, then the tangents per pair.

Pairs from 3 circles = ³C₂ = 3
Two separate circles (neither touching nor overlapping) have 4 common tangents: 2 direct + 2 transverse
Total = 3 × 4 = 12

For the maximum, place the circles so that no line touches all three; then none of the 12 is counted twice → option (d)

Why the others are wrong

• (a)
  17 — 17 exceeds 3 pairs × 4 tangents = 12. Two circles share at most 4 common tangents, and three circles form 3 pairs, so no arrangement reaches 17.
• (b)
  16 — 16 would need 4 pairs of circles, or more than 4 tangents per pair. Three circles give ³C₂ = 3 pairs, so the ceiling is 3 × 4 = 12.
• (c)
  9 — 9 is 3 tangents per pair, the count for two circles that touch externally. The stem says non-touching, so each pair has 4 tangents, giving 12.

How many common tangents two circles share depends on the distance d between their centres, compared with the radii r₁ ≥ r₂:

d > r₁ + r₂, separate circles: 4
d = r₁ + r₂, touching externally: 3
r₁ − r₂ < d < r₁ + r₂, intersecting: 2

d = r₁ − r₂, touching internally: 1
d < r₁ − r₂, one inside the other: 0

Key facts

• Two circles lying outside each other have 4 common tangents: 2 direct and 2 transverse.
• Circles touching externally have 3 common tangents, and circles touching internally have 1.
• Two intersecting circles have 2 common tangents, both direct.
• The number of pairs from n objects is ⁿC₂ = n(n − 1) ⁄ 2.

Common traps

• Counting 3 tangents per pair, the externally-touching case, and reaching 9 when the stem says non-touching.
• Stopping at 4, the count for one pair, and forgetting that three circles make three pairs.

No directly related past PYQ was found.