Multiple choice

The length of a direct common tangent and a transverse common tangent to two circles are 3√21 cm and 9 cm, respectively. If the radii of the two circles are in the ratio 1 ∶ 3, find the distance (in cm) between their centres.

  1. 15

  2. 14

  3. 18

  4. 16
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A Correct answer
Explanation

Let radii be r and 3r. Direct tangent length d = sqrt(D^2 - (3r-r)^2) = sqrt(D^2 - 4r^2) = 3*sqrt(21). Transverse tangent length t = sqrt(D^2 - (3r+r)^2) = sqrt(D^2 - 16r^2) = 9. Squaring: D^2 - 4r^2 = 189 and D^2 - 16r^2 = 81. Subtracting: 12r^2 = 108, so r^2 = 9, r = 3. Then D^2 - 4(9) = 189, D^2 = 225, D = 15.

AI explanation

Let the smaller radius be r and the larger radius be 3r based on the 1 to 3 ratio. Using the transverse common tangent formula, the square of the distance between centers equals the square of the tangent plus the square of the sum of the radii, so d squared equals 9 squared plus 4r squared. Using the direct common tangent formula, d squared also equals the square root of 21 quantity squared plus the square of the difference of the radii, giving 189 plus 4r squared, which shows consistency. Equating the two expressions for d squared to isolate r is not needed; simply use the given direct tangent length and radii difference to find d squared equals 189 plus 4r squared. Simultaneously solving with the transverse tangent reveals r is 3, making the radii 3 and 9. Substituting these radii into either tangent equation gives d squared equals 81 plus 144 for 225, meaning the distance is 15 centimeters.

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