Two circles touch internally. The sum of their areas is 116π cm² and distance between their centres is 6 cm. Find the radii of the circles.

Given: Sum of areas = 116π cm²

To find: Radii of the circles

Explanation:

Let “R” and “r” be the radii of the circles having centre “O” and “o”.

Then,

Sum of areas = 116π

⇒ πR² + πr² = 116π

⇒ R² + r² = 116 ….. (1)

Distance between their centres = 6 cm

⇒ Oo = 6 cm

⇒ R – r = 6 ….. (2)

Now,

(R+r)² + (R-r)² = 2(R² + r²)

⇒ (R+r)² + 6² = 2×116

⇒ (R+r)² + 36 = 232

⇒ (R+r)² = 232 – 36

⇒ (R+r)² = 196

⇒ R+r = 14 ….. (3)

Add (2) and (3)

⇒ R – r + R + r = 6 + 14

⇒ 2R = 20

⇒ R = 10 cm

Putting the value of R in (2) we get,

10 – r = 6

⇒ r = 4 cm

Hence the radii are 10 cm and 4 cm.