Multiple choice

If the length of a common internal tangent to two circles is $7$, and that of a common external tangent $11$, then the product of the radii of the two circles is:

  1. $18$
  2. $20$
  3. $16$
  4. $12$
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A Correct answer
Explanation

Let R and r be radii. Internal tangent length L_i^2 = d^2 - (R+r)^2 = 49. External tangent L_e^2 = d^2 - (R-r)^2 = 121. Subtracting gives (R+r)^2 - (R-r)^2 = 121 - 49 = 72. Thus, 4Rr = 72, so Rr = 18.

AI explanation

The lengths of the common external and internal tangents are related to the distance d between centers and the radii r1 and r2 by the identities L_ext squared equals d squared minus (r1 minus r2) squared and L_int squared equals d squared minus (r1 plus r2) squared. Subtracting the second equation from the first gives L_ext squared minus L_int squared equals 4r1r2. Substituting the given lengths 11 and 7, we calculate 121 minus 49 equals 72, meaning 4r1r2 equals 72 and the product r1r2 equals 18.

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