Two circles touch internally and their centres are O and O' as shown. The sum of their areas is 180 sq cm and the
distance between their centres is 6 cm. Find the diameter (in cm) of the larger circle.
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Two circles touch internally and their centres are O and O' as shown. The sum of their areas is 180 sq cm and the
distance between their centres is 6 cm. Find the diameter (in cm) of the larger circle.
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Given:
Two circles touch internally and their centres are O and O' as shown. The sum of their areas is 180 sq cm and the distance between their centres is 6 cm.
To find:
Find the diameter (in cm) of the larger circle.
Solution:
From given, we have,
The distance between the centres of circles is 6 cm.
Let the radius of the inner circle be 'a'
Let the radius of the outer circle be 'a + 6'
The sum of the areas of the circles is 180 sq cm.
πa² + π(a + 6)² = 180
πa² + πa² + 36 + 12πa = 180
2πa² + 12πa = 144
πa² + 6πa = 72
a² + 6a - 23 = 0
upon further solving, we get,
a = -3 ± 4√2
Therefore, the radius of the inner circle is, a = -3 ± 4√2
Therefore, the radius of the outer circle is, a + 6 = -3 ± 4√2 + 6 = 3 ± 4√2
The diameter of the larger circle is = 2 (3 ± 4√2)
Therefore, the diameter of the larger circle is 6 ± 8√2 cm
Answer:
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