If the length of a common external tangent to two circles is 9 and that of a common internal tangent is 5, then the product of the radii of the two circles is:
Show answer & explanation
Correct answer: C
Explanation
The correct answer is Option 3 i.e. 14.
We know that,
Length of common external tangent(Le) = √(d2 - (r1 + r2)2)...(1) and,
Length of common internal tangent(Li) = √(d2 - (r1 - r2)2)....(2)
(Le) = 9 and (Li) = 5 [given]
Solving eq 1
⇒ 9 = √(d2 - (r1 + r2)2)
⇒ 81 = d2 - (r1 + r2)2
⇒ d2 = 81 + (r1 + r2)2 ....(3)
Solving eq 2
⇒ 5 = √(d2 - (r1 - r2)2)
⇒ 25 = d2 - (r1 - r2)2
⇒ d2 = 25 + (r1 - r2)2 ....(4)
From eq 3 and 4
⇒ 81 + (r1 + r2)2 = 25 + (r1 - r2)2
⇒ (r1 + r2)2 - (r1 - r2)2 = 25 - 81
⇒ [(r1 + r2) - (r1 - r2)][(r1 + r2) + (r1 - r2)] = -56
⇒ 4r1r2 = -56
⇒ r1r2 = -14
Thus, the product of the radii of the two circles is 14