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)`

Answer 1

`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`