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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.
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Solution
Let r1, r2 and 01, 02 be the radii and angles subtended at the centre of two circles, respectively.
Let its radius = r1
l = r1θ1
= r1 `pi/3`
∴ r1 = `(3l)/pi` …(i)
For the second circle,
Let radius = r2
Arc length = l
The angle made by the arc at the centre, θ2 = 75°
= `75 xx π/180` radians
= `(5π)/12` radians
r2 = `(12l)/(5π)`
On dividing equation (i) by equation (ii)
`r^1/r^2 = (3l)/π + (12l)/(5π)`
= `(3l)/πxx(5π)/(12l)` = 5 : 4.
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