Question

Two circles touch externally. The sum of their areas is 130π sq. cm and the distance between their centres is 14 cm. Find the radii of the circles.

Answer 1

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
----------------
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

HOPE THIS WILL HELP YOU...

Answer 2

hi! here is your answer.

The radii of the circle are 11cm & 3cm or 3cm & 11cm.

hope this helps#!