Two circles, with radii of r₁ and r₂ respectively, have their centers separated by a distance of d. If the length of a direct common tangent is equal to the distance between the centers, which of the following is true?

Correct — C.
Direct common tangent: L² = d² − (r₁ − r₂)²
Put L = d: d² = d² − (r₁ − r₂)²

Cancel d² from both sides: (r₁ − r₂)² = 0
Take the square root: r₁ − r₂ = 0
So r₁ = r₂, two equal circles → option (c)

Why the others are wrong

• (a)
  d = r₁−r₂ — d = r₁ − r₂ makes the tangent length 0, not d: the formula becomes L² = d² − d². That is the case of two circles touching internally.
• (b)
  d = (r₁−r₂)² — Setting L = d leaves (r₁ − r₂)² = 0, not (r₁ − r₂)² = d. This option also equates a length with a squared length, so its units do not match.
• (d)
  r₁ = 2r₂ — Unequal radii make the tangent shorter than d. With r₁ = 2r₂, the difference r₁ − r₂ is r₂, so L² = d² − r₂², which is less than d².

For circles with centres d apart and radii r₁, r₂, the direct common tangent (both circles on the same side of it) has length √(d² − (r₁ − r₂)²). The transverse tangent, which crosses between them, has length √(d² − (r₁ + r₂)²).

Both come from one right triangle, made by drawing a line through the smaller circle's centre parallel to the tangent: the hypotenuse is d, one leg equals the tangent, the other is r₁ − r₂ (or r₁ + r₂ for the transverse).

So a direct tangent is never longer than d, and equals d exactly when the radii are equal.

Key facts

• Direct common tangent = √(d² − (r₁ − r₂)²)
• Transverse common tangent = √(d² − (r₁ + r₂)²)
• For equal circles the direct common tangent is parallel to the line of centres and equals d
• For circles touching externally (d = r₁ + r₂), the direct common tangent is 2√(r₁r₂)

Common traps

• Swapping the formulas: r₁ − r₂ goes with the direct tangent, r₁ + r₂ with the transverse one
• Picking d = r₁ − r₂ because it echoes the formula's r₁ − r₂, when that value makes the tangent length 0

No directly related past PYQ was found.