Question

If the arcs of the same length in two circles subtend angles ${65}^{\circ }and{110}^{\circ }$ at the centre, then the ratio of the radii of the circles is

• A. 22 : 13
• B. 11 : 13
• C. 22 : 15
• D. 21 : 13

Answer 1

The correct option is A

22 : 13

Let the angles substended 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:

${\theta }_1={65}^{\circ }=(65\times \frac{\pi }{180})\text{radian}{\theta }_2={65}^{\circ }=(110\times \frac{\pi }{180})\text{radian}{\theta }_1=\frac{1}{r_2}\Rightarrow r_1=\frac{1}{(65\times \frac{\pi }{180})}{\theta }_2=\frac{1}{r_2}\Rightarrow r_2=\frac{1}{110\times \frac{\pi }{180}}$

$\Rightarrow \frac{r_1}{r_2}=\frac{\frac{1}{(65\times \frac{\pi }{180})}}{\frac{1}{(110\times \frac{\pi }{180})}}=\frac{110}{65}=\frac{22}{13}\Rightarrow r_1:r_2=22:13$