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If the arcs of the same length in two circles $S_1$ and $S_2$ subtend angles $75^o $ and $120^o $ respectively at the centre. The ratio $\frac{{{S_1}}}{{{S_2}}}$ is equal to
$\frac{1}{5}$
$\frac{{81}}{{16}}$
$\frac{{64}}{{25}}$
$\frac{{25}}{{64}}$
Solution
$\theta=\frac{l}{r}$
$\theta \rightarrow$ angle (in radians)
$l \rightarrow$ are length
$r \rightarrow$ radius
$l_{1}=l_{2}=l(\operatorname{say})$
$\theta_{1}=75^{\circ} \quad ; \quad \theta_{2}=120^{\circ}$
$1^{0}=\left(\frac{\pi}{180}\right)^{c}$
$\theta_{1}=75^{\circ}=\left(\frac{5 \pi}{12}\right)^{\text {radians }}$
$\theta_{1}=120^{\circ}=\left(\frac{2 \pi}{3}\right)^{\text {radians }}$
$\frac{r_{1}}{r_{2}}=\frac{\frac{l_{1}}{\theta_{1}}}{\frac{\lambda_{2}}{\theta_{2}}}$
$=\frac{l_{1}}{\theta_{1}} \times \frac{\theta_{2}}{l_{2}}$
$r_{1}: r_{2}=8: 5$
