Two arcs of the same length subtend angles of $60^{o}$ and $75^{o}$ at the centres of the circles. What is the ratio of radii of two circles?

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Solution

$S_{1} = S, S_{2} = S$
$θ_{1} = 60^{o}, θ_{2} = 75^{o}$
$S_{1} = r_{1} θ_{1}, S_{2} = r_{2} θ_{2}$
$S = r_{1} θ_{1}, S = r_{2} θ_{1}$
$∴ r_{1} θ_{1} = r_{2} θ_{2}$
$∴ \frac{r_{1}}{r_{2}} = \frac{θ_{2}}{θ_{1}} = \frac{75}{60} = \frac{5}{4}$
$∴$ ratio of radii of two circles is $5 : 4$

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Two arcs of the same length subtend angles of 60o and 75o at the centres of the circles. What is the ratio of radii of two circles?

S1=S, S2=Sθ1=60o, θ2=75oS1=r1 θ1, S2=r2 θ2S=r1 θ1, S=r2 θ1∴ r1 θ1=r2 θ2∴ r1r2=θ2θ1=7560=54∴ ratio of radii of two circles is 5:4

Two arcs of the same length subtend angles of $60^{o}$ and $75^{o}$ at the centres of the circles. What is the ratio of radii of two circles?