Question

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

• A. $22:13$
• B. $11:13$
• C. $10:51$
• D. None of these

Answer 1

Answer: (A)

Let $r_1$ and $r_2$ be the radii of the two circles.
Given that, ${\theta }_1=65^{\circ }=\frac{\pi }{180}\times 65=\frac{13\pi }{36}$ radian
and ${\theta }_2=110^{\circ }=\frac{\pi }{180}\times 110=\frac{22\pi }{36}$ radian
Let $l$ be the length of each of the arc.
Then, $l=r_1{\theta }_1=r_2{\theta }_2$, which gives
$\frac{13\pi }{36}\times r_1=\frac{22\pi }{36}\times r_2$
$\text{, i.e. },\frac{r_1}{r_2}=\frac{22}{13}$
Hence, $r_1:r_2=22:13$.