Mensuration-34

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(a) Two circles touch externally. The sum of their areas is 130 Π sq cm and the distance between their centers is 14 cm.
Find the radii of the circles.

(b) Two circles touch internally. The sum of their areas is 116 Π sq cm and the distance between their centers is 6 cm.
Find the radii of the circles.

Solution:(a) Since the two circles touch each other externally so, the distance between their centers = sum of their radii.

Let the center of one circle be x cm. Then the radius of the other circles is (14 - x) cm.

Therefore, Sum of their areas = Π x2 + Π (14 - x)2 = 130 Π (given)

=> x2 + 196 - 28 x + x2 = 130;

=> x2 - 14 x + 33 = 0

=> (x - 3) (x - 11) = 0

=> x = 3 or 11

Therefore, 14 - x = 11 or 3

Hence, the radii of the two circles are 11 cm and 3 cm respectively.

(b) Since the two circles touch each other internally, so the distance between their centers = difference of their radii

Let the radius of the larger circles be x cm

Then, radius of the smaller circle = (x - 6) cm

Therefore, Sum of their areas = Π r2 + Π (x - 6)2 = 116 Π (given)

=> x2 + x2 - 12x + 36 = 116

=> x2 - 6x - 40 = 0

=> (x - 10) (x + 4) = 0

=> x = - 4, 10.

Rejecting negative values, x = 10

Therefore, the radii of the two circles are 10 cm and 4 cm

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