Question

Two circles touch internally and their centres are 0 and O' as shown.
The sum of their areas is 180 sq. cm. and the distance between their
centres is 6 cm. What is the diameter of the larger circle?
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Answer 1

Answer:

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