Two circles with radii r₁ and r₂ touch each other externally. If the length of their direct common tangent is T, which of the following is the correct relationship between T, r₁, and r₂?

Correct — C. Start from the direct-tangent formula and apply the touching condition.

Direct tangent: T² = d² − (r₁ − r₂)²
Touching externally: d = r₁ + r₂
T² = (r₁ + r₂)² − (r₁ − r₂)²

Expand: (r₁² + 2r₁r₂ + r₂²) − (r₁² − 2r₁r₂ + r₂²) = 4r₁r₂
T = √(4r₁r₂) = 2√(r₁r₂) → option (c)

Why the others are wrong

• (a)
  T = r₁ + r₂ — r₁ + r₂ is the distance between the centres, not the tangent. With r₁ = 4, r₂ = 1: d = 5 but T = √(5² − 3²) = 4. The two agree only when r₁ = r₂.
• (b)
  T = 2(r₁ + r₂) — 2(r₁ + r₂) is twice the centre distance, but T² = d² − (r₁ − r₂)² means T can never exceed d. With r₁ = 4, r₂ = 1 it gives 10 against the true 4.
• (d)
  T = r₁² + r₂² — r₁² + r₂² is in square units, so it cannot be a length. With r₁ = 4, r₂ = 1 it gives 17 against the true tangent of 4.

Direct common tangent, from one right triangle. Both radii to the points of contact are perpendicular to the tangent, so they are parallel.

A line from the smaller centre, parallel to the tangent, meets the larger radius and forms a right triangle: hypotenuse d, legs r₁ − r₂ and T.

So T² = d² − (r₁ − r₂)². Touching externally, d = r₁ + r₂ and it becomes T = 2√(r₁r₂).

Key facts

• Direct common tangent: T = √(d² − (r₁ − r₂)²).
• Transverse common tangent: √(d² − (r₁ + r₂)²), for circles lying apart.
• Circles touching externally: direct common tangent = 2√(r₁r₂).

Common traps

• Putting r₁ + r₂ inside the direct formula — that is the transverse formula, and at d = r₁ + r₂ it returns 0.
• Stopping at r₁ + r₂, the centre distance: it equals the tangent only for equal radii.

No directly related past PYQ was found.