Question

Two circles touch internally. The sum of their areas is116 cm² and the distance between their centres is 6cm. Find the radii of the circles.

Answer 1

π×r²+π×R² = 116
22/7(r²+R²) = 116
r²+R² = (58×7)÷11
r+R = 6
r = 6-R
36+R²-12R+R² = 406/11
R²-6R+18 = 203/11
11R²-66R+198-203 = 0
11R²-66R-5 = 0
R = (66±√4136)÷22
= (33±√1034)÷11
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Answer 2

Answer:

Step-by-step explanation:

Let a circle with center O And radius R.

let

another circle inside the first circle  with center o' and radius r .

A/Q,

Area of 1st circle + area of 2nd circle = 116π cm²

⇒ πR² + πr² = 116π 

⇒ π(R² + r²)  = 116π

⇒ R² + r² =116 --------------------(i)

Now,

Distance between the centers of circles = 6 cm

i.e, R - r = 6

⇒  R = r + 6  -------------------(ii)

From Eqn (i) & (ii),

(r + 6)² + r² = 116

⇒ r² + 12r +36 + r² =116

⇒ 2r² +12r +36 -116 = 0

⇒ 2r² +12r - 80 = 0

⇒ r² +6r - 40 = 0

⇒ r² +10r - 4r - 40 = 0

⇒ r(r + 10) - 4(r + 10) = 0

⇒ (r + 10)(r - 4) = 0

 hence  r = 4 cm

r ≠ -10 cm        {∵ length can't be -ve}

Therefore radii of the circles are

r = 4 cm ,

R = 4 + 6 = 10 cm.