Question

If the arcs of the same lengths in two circles subtend angles ${65}^o$ and ${110}^o$ at the center, find the ratio of their radii.

• A. $\frac{11}{13}$
• B. $\frac{22}{13}$
• C. $\frac{13}{11}$
• D. $\frac{11}{26}$

Answer 1

The correct option is B $\frac{22}{13}$

Let, $r_1$ and $r_2$ be the radii of the two circles.
Given, ${\theta }_1={65}^o\Rightarrow \frac{\pi }{180}\times {65}^o=\frac{13\pi }{36}radian$

and ${\theta }_2={110}^o\Rightarrow \frac{\pi }{180}\times {110}^o=\frac{22\pi }{36}radian$

Let, $l$ be the length of each of the arc. Then,
$l=r_1{\theta }_1=r_2{\theta }_2$

so, $\frac{13\pi }{36}\times r_1=\frac{22\pi }{36}\times r_2\frac{r_1}{r_2}=\frac{22\pi }{36}\times \frac{36}{13\pi }=\frac{22}{13}$

Hence, $\frac{r_1}{r_2}=\frac{22}{13}$