The centres of two circles are 36 cm apart. If the radii of these two circles are 15 cm and 9 cm, respectively, then what is the sum of the lengths (in cm) of a direct common tangent and a transverse common tangent of these two circles?

Correct — D. Two standard lengths, then factor the sum into the form the options are printed in.

Direct common tangent = √(d² − (r₁ − r₂)²)
= √(36² − 6²) = √(1296 − 36) = √1260 = 6√35

Transverse common tangent = √(d² − (r₁ + r₂)²)
= √(36² − 24²) = √(1296 − 576) = √720 = 12√5

Sum = 6√35 + 12√5
√35 = √5 × √7, so 6√35 = 6√5 × √7 and 12√5 = 6√5 × 2
Sum = 6√5(√7 + 2), about 62.3 cm

That is option (d).

Why the others are wrong

• (a)
  Option (a), 6√7(√7 + 2), expands to 42 + 12√7. Its second term puts a 7 under the root of the transverse tangent, but √720 = 144 × 5 under the root, giving 12√5.
• (b)
  Option (b), 6√5(√5 + 2), has first term 6√5 × √5 = 30. That would make the direct tangent √900, not the √1260 that 36² − 6² actually gives.
• (c)
  Option (c), 6√7(√5 + 2), gets the direct tangent right at 6√35 but pairs it with 12√7. The transverse tangent squares to 1296 − 576 = 720, whose root is 12√5.

Both lengths come out of one right triangle. Slide the tangent parallel to itself onto the line of centres and you are left with a right triangle whose hypotenuse is d, the distance between the centres.

For a direct common tangent the remaining side is the difference of the radii, so the length is √(d² − (r₁ − r₂)²).

For a transverse common tangent the line crosses between the circles, so the remaining side is the sum of the radii: √(d² − (r₁ + r₂)²).

A transverse tangent exists only when d > r₁ + r₂. Here 36 > 24, so both tangents are real.

Key facts

• Direct common tangent = √(d² − (r₁ − r₂)²), here √(1296 − 36) = √1260 = 6√35.
• Transverse common tangent = √(d² − (r₁ + r₂)²), here √(1296 − 576) = √720 = 12√5.
• √35 = √5 × √7, so 6√35 + 12√5 factors as 6√5(√7 + 2).
• A transverse common tangent exists only when the centres are farther apart than the sum of the radii.

Common traps

• Swapping the formulas and using the sum of the radii for the direct tangent.
• Stopping at 6√35 + 12√5, a correct value that appears in none of the printed options.
• Glancing past option (a) and option (d) as though they were the same, when only the outer root differs.

No directly related past PYQ was found.