The radii of two circles are 6 cm and 2 cm. The centers of the two circles are 10 cm apart. What is the length of a transverse common tangent?

Correct — A. A transverse tangent crosses between the circles, so its formula uses the sum of the radii.

Formula: L = √(d² − (r₁ + r₂)²)
Sum of radii: 6 + 2 = 8 cm
Subtract squares: 10² − 8² = 100 − 64 = 36
Root: L = √36 = 6 cm → option (a)

Why the others are wrong

• (b)
  8 cm — 8 cm is r₁ + r₂, the leg you subtract, not the tangent. It goes into the formula squared: 10² − 8² = 36, so the tangent is √36 = 6.
• (c)
  10 cm — 10 cm is the distance between the centres, the hypotenuse of the right triangle. The tangent is a leg of that triangle, so it must be shorter than 10.
• (d)
  12 cm — 12 cm is longer than the 10 cm between the centres. The tangent is a leg of a right triangle whose hypotenuse is 10 cm, so it can never reach 12.

Radii drawn to the points of contact are perpendicular to the tangent. For a transverse tangent they point in opposite directions.

Slide the tangent parallel to itself until it passes through the smaller circle's centre. It now forms a right triangle: hypotenuse d, one leg r₁ + r₂, and the tangent length as the remaining leg.

For a direct tangent the radii point the same way, so that leg becomes r₁ − r₂.

Key facts

• Transverse common tangent: L = √(d² − (r₁ + r₂)²).
• Direct common tangent: L = √(d² − (r₁ − r₂)²).
• Two separate circles (d > r₁ + r₂) have four common tangents: two direct and two transverse.
• For the same two circles, a transverse tangent is always shorter than a direct tangent.

Common traps

• Using r₁ − r₂ for a transverse tangent. That is the direct-tangent formula and gives √84 ≈ 9.17 cm here.
• Subtracting before squaring: 10 − 8 = 2. Square first, subtract, then take the root.

No directly related past PYQ was found.