Two circles have radii of 12 cm and 4 cm. If the length of a direct common tangent is 15 cm, what is the distance between their centers?

Correct — B. The direct common tangent, the line of centres and the difference of the radii form a right triangle.

Formula: L² = d² − (r₁ − r₂)²
Difference of radii: 12 − 4 = 8 cm
Substitute: 15² = d² − 8², so d² = 225 + 64 = 289
Square root: d = 17 cm → option (b)

Why the others are wrong

• (a)
  15 cm — 15 cm is the tangent length itself. The centre distance is the hypotenuse of a right triangle with legs 15 and 8, so it must be longer than 15.
• (c)
  18 cm — 18 cm gives a direct tangent of √(18² − 8²) = √260 ≈ 16.1 cm, not the 15 cm in the question.
• (d)
  20 cm — 20 cm gives a direct tangent of √(20² − 8²) = √336 ≈ 18.3 cm, longer than the given 15 cm.

Radii drawn to the two points of contact are both perpendicular to the tangent, so they are parallel.

Through the smaller centre, draw a line parallel to the tangent. It forms a right triangle with legs L and r₁ − r₂ and hypotenuse d, so L² = d² − (r₁ − r₂)².

For the transverse tangent, which crosses between the circles, the radii add: M² = d² − (r₁ + r₂)².

Key facts

• Direct common tangent: L = √(d² − (r₁ − r₂)²).
• Transverse common tangent: M = √(d² − (r₁ + r₂)²).
• A radius drawn to the point of contact is perpendicular to the tangent.

Common traps

• Adding the radii (12 + 4 = 16) for a direct tangent. The sum belongs to the transverse tangent and would give d = √481 ≈ 21.9 cm.
• Picking 15 cm, the tangent length, as the centre distance.

No directly related past PYQ was found.