The circumference of a circle is equal to the sum of the circumferences of two circles with radii 10 cm and 12 cm. The radius of the circle formed is :

1. 2 cm

2. 44 cm

3. 22 cm

4. 244 cm

Given:

The circumference of a circle = The sum of the circumferences of two circles with radii 10 cm and 12 cm.

As we know, the circumference of a circle = 2πr

For r = 10 cm:

Circumference of the circle = 2 x $\dfrac{22}{7}$ x 10

= $\dfrac{44}{7}$ x 10

= $\dfrac{440}{7}$

For r = 12 cm

Circumference of the circle = 2 x $\dfrac{22}{7}$ x 12

= $\dfrac{44}{7}$ x 12

= $\dfrac{528}{7}$

Let R be the radius of the larger circle.

The circumference of the larger circle =

$\dfrac{440}{7} + \dfrac{528}{7}\\[1em] ⇒ 2πR = \dfrac{440 + 528}{7}\\[1em] ⇒ 2 \times \dfrac{22}{7} \times R = \dfrac{440 + 528}{7}\\[1em] ⇒ \dfrac{44}{7} \times R = \dfrac{968}{7}\\[1em] ⇒ R = \dfrac{968 \times 7}{7 \times 44}\\[1em] ⇒ R = \dfrac{968 \times \cancel{7}}{\cancel{7} \times 44}\\[1em] ⇒ R = \dfrac{968}{44}\\[1em] ⇒ R = 22 \text{ cm}$

Hence, option 3 is the correct option.