Hello, snigdha!
Another approach . . .
Two circles touch internally.
The sum of their areas is 116π cm².
The distance between their centres is 6 cm.
Find the radii of the circles. Code:
* * *
* *
* *
* *
* *
- * o * -
: * | * :
: * 6
R * * | * * :
: * o * -
: * * | * * r
- * * * -
The radius of the large circle is $\displaystyle R.$
The radius of the small circle is $\displaystyle r.$
We see that: .$\displaystyle R \:=\:r+6$
The area of the large circle is: .$\displaystyle A_1 \:=\:\pi R^2 \:=\:\pi(r+6)^2 $
The area of the small circle is: .$\displaystyle A_2 \:=\:\pi r^2$
The sum of the areas is $\displaystyle 116\pi$
So we have: .$\displaystyle \pi(r+6)^2 + \pi r^2 \:=\:116\pi$
. . Now solve for $\displaystyle r.$