Question

If the arcs of the same length in two circles subtend angles 65° and 110° at the centre, than the ratio of the radii of the circles is

Options

• 22 : 13

• 11 : 13

• 22 : 15

• 21 : 13

Answer 1

22:13
Let the angles subtended at the centres by the arcs and radii of the first and second circles be \[\theta_1\text{ and } r_1\text{ and }\theta_2\text{ and }r_2 ,\] respectively.
We have:
\[\theta_1 = 65^\circ = \left( 65 \times \frac{\pi}{180} \right)\text{ radian }\]

\[\theta_2 = 65^\circ = \left( 110 \times \frac{\pi}{180} \right)\text{ radian }\]

\[\theta_1 = \frac{l}{r_1}\]

\[\Rightarrow r_1 = \frac{l}{\left( 65 \times \frac{\pi}{180} \right)}\]

\[\theta_2 = \frac{l}{r_2}\]

\[\Rightarrow r_2 = \frac{l}{\left( 110 \times \frac{\pi}{180} \right)}\]
\[\Rightarrow \frac{r_1}{r_2} = \frac{\frac{l}{\left( 65 \times \frac{\pi}{180} \right)}}{\frac{l}{\left( 110 \times \frac{\pi}{180} \right)}} = \frac{110}{65} = \frac{22}{13}\]

\[\Rightarrow r_1 : r_2 = 22: 13\]