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

$\begin{array}{1 1}(A)\;5 : 4&(B)\;4 : 5\\(C)\;5 : 3&(D)\;6 : 5\end{array}$

Toolbox:
• $\theta=\large\frac{l}{r}$
• $\theta$=angle subtended by arc
• $l$=length of the arc
• $r$=radius of the circle
• $1^{\large\circ}=(\large\frac{\pi}{180})$ radian=0.01746(approx)
Let the length of the circle be $l$
Angle of the circle 1 =$60^{\large\circ}$
Angle of the circle 2 =$75^{\large\circ}$
Let the radius be $r_1$ and $r_2$
$l=r_1\theta$
$\Rightarrow r_1\times 60\times \large\frac{\pi}{180}$
$\Rightarrow \large\frac{\pi r_1}{3}$
$l=r_2\theta$
$\Rightarrow r_2\times 75\times \large\frac{\pi}{180}$
$\Rightarrow \large\frac{5\pi r_2}{12}$
Since $l$ is same for both the circles
$\Rightarrow \large\frac{\pi r_1}{3}=\frac{5\pi r_2}{12}$
$\Rightarrow r_1 : r_2=5 : 4$
Hence (A) is the correct answer.