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Let AB be a sector of a circle with centre O and radius d. $∠ A O B = θ < π 2$ , and D be a point on OA such that BD is perpendicular to OA. Let E be the midpoint of BD and F be a point on the arc AB such that EF is parallel to OA. Then the ratio of length of the arc AF to the length of the arc AB is?

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12

θ 2

12 sin θ

sin − 1 12 sin θ

answer is D.

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Detailed Solution

Let the diagram of the situation be,
Question Image

Let

∠ A O F = α

. If s is the arc length and r is the radius of the circle, then the length of the arc of a circle is given as

s = r θ

. So,

l (A F) l (A B) = d ⋅ α d ⋅ θ = α θ

From the figure, we get

Δ D O C

and

Δ E F C

are right triangles. So,

∠ O D C = ∠ F E C = 90 ∘

. Also,

∠ D O C = ∠ E F C = α

as they are alternate angles. In

Δ B D O

sin θ = B D O B B D = d sin θ

As E is the midpoint of BD, we get

E D = 12 d sin θ

. So,

O C (sin α) + C F (sin α)

.
We know that,

sin α = C D O C

and

sin α = C E C F

O C (sin α) + C F (sin α) = O C C D O C + C F C E C F = C D + C E = E D

We know that,

C F = d − O C

and

E D = 12 d sin θ

. So,

O C (sin α) + (d − O C) (sin α) = 12 d sin θ d sin α = 12 d sin θ sin α = 12 sin θ α = sin − 1 12 sin θ

Hence, the correct option is 4.

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Let AB be a sector of a circle with centre O and radius d. ∠AOB=θ < π 2 , and D be a point on OA such that BD is perpendicular to OA. Let E be the midpoint of BD and F be a point on the arc AB such that EF is parallel to OA. Then the ratio of length of the arc AF to the length of the arc AB is?

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