2 circles touch each other externally and the sum of their areas is 52 cm2.if the distance between the centres is 10 cm,find their radii.
Answers
SOLUTION :
Let r1 & r2 be the Radii of the two circles having centres A & B.
Given:
Sum of areas of two circles= 130π cm²
Distance between their centres = 14 cm
Area of circle = πr²
πr1² + πr2² = 130π
π(r1² + r2²) = 130π
r1² + r2² = 130…………(1)
Distance between their centres = 14 cm
r1 + r2 = 14 cm …………..(2)
On squaring both sides,
(r1 + r2)² = (14)²
(r1² + r2²)+2r1r2 = 196
[ (a+b)²= a²+b²+2ab]
130 + 2r1r2 = 196
2r1r2 = 196 - 130
2r1r2 = 66
r1r2 = 66/2= 33
r1r2 = 33………..(3)
(r1 - r2)² = (r1 + r2)² - 4r1r2
[(a-b)² = (a+b)² - 4ab]
(r1 - r2)² = 14² - 4(33)
[From eq 2 & 3]
(r1 - r2)² = 196 - 132
(r1 - r2)² = 64
(r1 - r2) = √64 = 8
(r1 - r2) = 8……………..(4)
On Solving eq 2 & eq 4
r1 + r2 = 14
r1 - r2 = 8
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2r1 = 22
r1 = 22/2= 11
r1 = 11cm
On Putting the value of r1 in eq 4
r1 - r2 = 8
11 - r2 = 8
11 -8 = r2
r2 = 3 cm
Hence, the Radii of the circles be 11 cm & 3 cm
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