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
(a) 22 : 13
(b) 11 : 13
(c) 22 : 15
(d) 21 : 13

Answer 1

(a) 22:13

Let the angles subtended at the centres by the arcs and radii of the first and second circles be ${\theta }_1\mathrm{and}{r}_1\mathrm{and}{\theta }_2\mathrm{and}{r}_2,$ respectively.
We have:
${\theta }_1=65°=\left(65\times {\displaystyle \frac{\mathrm{\pi }}{180}}\right)\mathrm{radian}$
${\theta }_2=65°=\left(110\times {\displaystyle \frac{\mathrm{\pi }}{180}}\right)\mathrm{radian}$
${\mathrm{\theta }}_1=\frac{l}{r_1}$
$\Rightarrow r_1=\frac{l}{\left(65\times {\displaystyle \frac{\mathrm{\pi }}{180}}\right)}$

${\mathrm{\theta }}_2=\frac{l}{r_2}$
$\Rightarrow r_2=\frac{l}{\left(110\times {\displaystyle \frac{\mathrm{\pi }}{180}}\right)}$

$\Rightarrow \frac{r_1}{r_2}=\frac{\frac{l}{\left(65\times {\displaystyle \frac{\mathrm{\pi }}{180}}\right)}}{\frac{l}{\left(110\times {\displaystyle \frac{\mathrm{\pi }}{180}}\right)}}=\frac{110}{65}=\frac{22}{13}$
$\Rightarrow r_1:r_2=22:13$