#### Question

Two circles touch internally. The sum of their areas is 116 π cm^{2} and the distance between their centres is 6 cm. Find the radii of the circles ?

#### Solution

Let *r*_{1} be the radius of the bigger circle and *r*_{2} be the radius of the smaller circle.

It is given that the two circles touch each other internally.

∴ Difference between their radii = Distance between the centres of the two circles

⇒ *r*_{1 }− *r*_{2} = 6 cm .....(1)

Also,

Sum of their areas = 116π cm^{2 }

\[\therefore \pi {r_1}^2 + \pi {r_2}^2 = 116\pi\]

\[ \Rightarrow {r_1}^2 + {r_2}^2 = 116 . . . . . \left( 2 \right)\]

From (1) and (2), we have

\[\left( r_2 + 6 \right)^2 + {r_2}^2 = 116\]

\[ \Rightarrow {r_2}^2 + 12 r_2 + 36 + {r_2}^2 = 116\]

\[ \Rightarrow 2 {r_2}^2 + 12 r_2 - 80 = 0\]

\[ \Rightarrow {r_2}^2 + 6 r_2 - 40 = 0\]

\[ \Rightarrow {r_2}^2 + 10 r_2 - 4 r_2 - 40 = 0\]

\[ \Rightarrow r_2 \left( r_2 + 10 \right) - 4\left( r_2 + 10 \right) = 0\]

\[ \Rightarrow \left( r_2 + 10 \right)\left( r_2 - 4 \right) = 0\]

\[ \Rightarrow r_2 + 10 = 0 or r_2 - 4 = 0\]

\[ \Rightarrow r_2 = - 10 or r_2 = 4\]

Since the radius of a circle cannot be negative, so *r*_{2} = 4 cm.

∴ *r*_{1 }= *r*_{2} + 6 = 4 + 6 = 10 cm

Thus, the radii of the circles are 4 cm and 10 cm.