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Question
Choose the correct alternative:
A pair of tangents are drawn to a unit circle with center at the origin and these tangents intersect at A enclosing an angle of 60°. The area enclosed by these tangents and the area of the circle is
Options
`2/sqrt(3) - pi/6`
`sqrt(3) - pi/3`
`pi/3 - sqrt(3)/6`
`sqrt(3)(1 - pi/6)`
Solution
`sqrt(3) - pi/3`
Explanation;
In Δ OAP,
sin 30° = `1/"OP"`
∴ OP = 2
cos 30° = `"AP"/"OP"`
∴ `sqrt3/2 = "AP"/2`
∴ AP = `sqrt3`
A(`square`AOBP) = 2A(Δ OAP)
`= 2 xx 1/2 xx 1 xx sqrt3 = sqrt3`
A(sector AOB) = `1/2 xx (1)^2 xx (2pi)/3 = pi/3`
Required area = A( AOBP) - A(sector AOB)
`= sqrt3 - pi/3`
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