Two circles C 1 and C 2 do not intersect. The radii of C 1 and C 2 are 3 cm…
2019
Two circles C 1 and C 2 do not intersect. The radii of C 1 and C 2 are 3 cm and 2 cm respectively and the distance between their centres is 6 cm. The direct common tangents meet at S 1 . Find O 2 S 1 .
- A.
13 cm
- B.
10 cm
- C.
11 cm
- D.
12 cm
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Correct answer: D
Given: radii are 3 cm and 2 cm, and the distance between centers O1 and O2 is 6 cm.
Key insight: The intersection point of the direct common tangents is the external center of homothety of the two circles. This point lies on the line joining the centres and its distances to the centres are in the ratio of the radii (3:2).
Let S be the intersection of the direct common tangents. Place O1 at x = 0 and O2 at x = 6 on a number line, and let S be at x = t.
By the homothety property, O1S : O2S = 3 : 2, so t/(t-6) = 3/2.
Solve the equation: 2t = 3(t - 6) ⇒ 2t = 3t - 18 ⇒ t = 18.
Therefore O2S = t - 6 = 18 - 6 = 12 cm.
Answer: 12 cm