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Question

If the arcs of the same length in two circles substend angles 65 and 110 at the centre, find the ratio of their radii.

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Solution

Let, C1 and C2 are two circles with same Arc length l.

That is AB = CD = l

Let, θ1 and θ2 are two angles subtended by arc AB and CD on respective circles.

Let, OA = OB = r [radius of C1]

and OC = OA = R [radius of C2]

Also,

θ1=65=(65π180)cand θ2=110=(110π180)c

We know

θ=arcradius For C1 θ1=ABr
θ1=lr . . . (i)

r=1θ1

For C2

θ2=CDR θ2=lR
R=lθ2 . . .(ii)

From (i) and (ii)

rR=lθ1lθ2=θ2θ1=110π18065π180=2213

r : R = 22 : 13


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