If, in two circles, arcs of the same length subtend angles $60^{0}$ and $75^{0}$ at the centre, find the ratio of their radii.

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Solution

Let $r_{1}$ and $r_{2}$ be radii of two circles in which arcs of same length l subtend angles $θ_{1} = 60^{0}$ and $θ_{2} = 75^{0}$ respectively.

$∴ θ_{1} = \frac{l}{r_{1}} \Rightarrow {(60 \times \frac{π}{180})}^{C} = \frac{l}{r_{1}}$

$\Rightarrow r_{1} = \frac{3 l}{π}$ ...(i)

Also, $θ_{2} = \frac{l}{r_{2}} \Rightarrow 75 \times \frac{π}{180} = \frac{l}{r_{2}}$

$\Rightarrow r_{2} = \frac{l 2 l}{5 π}$ ...(ii)

From (i) and (ii)

$\frac{r_{1}}{r_{2}} = \frac{\frac{3 l}{π}}{\frac{12 l}{5 π}} \Rightarrow \frac{r_{1}}{r_{2}} = \frac{3 l}{π} \times \frac{5 π}{12 l} = \frac{5}{4}$

Hence $r_{1} : r_{2}$ = 5:4.

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