Two circle touch internally. The sum of their areas is 110 cm² and the distance between there centers is 6cm find the radius of circles.

The figure is given below:

The diagram above shows two circles with centers A and B touching internally at a point.

Let the radius of Circle with Centre A = a and the radius of Circle with center B be b.

It is Given that a – b = 6 ..(i)

And Also the sum of the areas of two circles = 110

Therefore, π a² + π b² = 110………(ii)

Now, a² + b² = 35 …….(iii)

Now we have to Solve (i) and (iii)

From equation (i), a = b + 6

Putting this in equation (iii)

(b + 6)² + b² = 35

2 b² + 12 b + 36 = 35

2 b² + 12 b + 1 = 0

Solving this quadratic equation we get,

Putting this value in (i) we get,

A = b + 6