Question

If the arcs of the same length in two circles substend angles ${65}^{\circ }and{110}^{\circ }$ at the centre, find the ratio of their radii.

Answer 1

Let, $C_{1}and{C}_2$ are two circles with same Arc length l.

That is AB = CD = l

Let, ${\theta }_{1}and{\theta }_2$ are two angles subtended by arc AB and CD on respective circles.

Let, OA = OB = r [radius of $C_1$]

and OC = OA = R [radius of $C_2$]

Also,

${\theta }_1={65}^{\circ }={\left(\frac{65\pi }{180}\right)}^{c}and{\theta }_2={110}^{\circ }={\left(\frac{110\pi }{180}\right)}^c$

We know

$\theta =\frac{\text{arc}}{\text{radius}}\therefore For{C}_1\Rightarrow {\theta }_1=\frac{AB}{r}$
$\Rightarrow {\theta }_1=\frac{l}{r}$ . . . (i)

$\Rightarrow r=\frac{1}{{\theta }_1}$

For $C_2$

${\theta }_2=\frac{CD}{R}\Rightarrow {\theta }_2=\frac{l}{R}$
$\Rightarrow R=\frac{l}{{\theta }_2}$ . . .(ii)

From (i) and (ii)

$\frac{r}{R}=\frac{\frac{l}{{\theta }_1}}{\frac{l}{{\theta }_2}}=\frac{{\theta }_2}{{\theta }_1}=\frac{\frac{110\pi }{180}}{\frac{65\pi }{180}}=\frac{22}{13}$

r : R = 22 : 13