Mathematics
Two circles touch each other externally. The sum of their areas is 74π cm2 and the distance between their centres is 12 cm. Find the diameters of the circle.
Mensuration
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Answer
Given:
Sum of area = 74π cm2
The distance between their centres = 12 cm
Let r1 and r2 be the radius of 2 circles.
⇒ A1 + A2 = 74π
⇒ πr12 + πr22 = 74π
⇒ π(r12 + r22) = 74π
⇒ (r12 + r22) = 74
⇒ r12 + r22 = 74 ……………(1)
And, r1 + r2 = 12
⇒ r1 = 12 - r2
Putting the value of r1 in equation (1),
⇒ (12 - r2)2 + r22 = 74
⇒ 122 + r22 - 2 x 12 x r2 + r22 = 74
⇒ 144 + 2r22 - 24r2 = 74
⇒ 144 + 2r22 - 24r2 - 74 = 0
⇒ 2r22 - 24r2 + 70 = 0
⇒ r22 - 12r2 + 35 = 0
⇒ r22 - 7r2 - 5r2 + 35 = 0
⇒ r2(r2 - 7) - 5(r2 - 7) = 0
⇒ (r2 - 7)(r2 - 5) = 0
⇒ r2 = 7 cm or 5 cm
⇒ r1 = 12 - r2 = 12 - 7 or 12 - 5 = 5 cm or 7 cm
Hence, the radius of 2 circles = 7 cm and 5 cm.
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