Two circles touch internally. The sum of their areas is 116π cm2, and the…
2023
Two circles touch internally. The sum of their areas is 116π cm2, and the distance between their centres is 6 cm. The radii of the circles are respectively:
Answer: B. 10 cm and 4 cm — Concept: For two circles that touch internally, the distance between their centres equals the difference of their radii. The area of a circle of radius r is…
- A.
16 cm and 10 cm
- B.
10 cm and 4 cm
- C.
12 cm and 6 cm
- D.
More than one of the above
- E.
None of the above
Attempted by 1 students.
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Correct answer: B
Concept: For two circles that touch internally, the distance between their centres equals the difference of their radii.
The area of a circle of radius r is πr2, so the sum of the areas gives an equation in the squares of the two radii.
Application:
Let the larger radius be R cm and the smaller radius be r cm, where R > r.
Internal tangency gives R − r = 6, hence R = r + 6.
From the area sum, π(R2 + r2) = 116π, so R2 + r2 = 116.
Substitute R = r + 6: (r + 6)2 + r2 = 116.
Expand and simplify: 2r2 + 12r − 80 = 0, hence r2 + 6r − 40 = 0.
Factor: (r + 10)(r − 4) = 0. A radius is positive, so r = 4 cm and R = 10 cm.
Cross-check:
The radius difference is 10 − 4 = 6 cm, which equals the centre distance.
The area sum is π(102 + 42) = π(100 + 16) = 116π cm2.
Therefore, the radii are 10 cm and 4 cm.