Let AB be a sector of a circle with centre O and radius d, \angle\AOB = \theta(

Correct Answer is : sin−1(12sinθ)θ

{\sin^{-1}(1/2\sin\theta)}/\theta

KVPY SX SB Exam 2016 Question Paper

Question : 85 of 120

Marks: +1, -0

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 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 -

$1/2$
$θ/2$
$1/2\sinθ$
${\sin^{-1}(1/2\sinθ)}/θ$

Validate

Solution:

CF = r(1 – cosθ sec ϕ)
EC = r(sinϕ – cosθ tan ϕ)
CD = r cosθ tan ϕ
EC + CD = ED
r sin ϕ

= {\r\sinθ}/2

\ϕ=\sin^{-1} ({\sinθ}/2)

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