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Question

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

Solution
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Let the radii of the two circles be r1 and r2. Let an arc of length I subtend an angle of 60 at the centre of the circle of radius r1, while let an arc of length I subtend an angle of 75 at the centre of the circle of radius r2.
Now, 60=π3 radian and 75=5π12 radian
We know that in a circle of radius r unit, if an arc of length l unit subtends an angle θ radian at the centre, then
θ=lr or l=rθ
l=r1π3 and l=r25π12
r1π3=r25π12
r1=r254
r1r2=54

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