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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
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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