KVPY SX SB Exam 2013 Question Paper

Question : 85 of 120

Marks: +1, -0

Let XY be the diameter of a semicircle with centre O . Let be a variable point on the semicircle and B another point on the semicircle such that AB is parallel to XY. The value of ∠BOY for which the inradius of triangle AOB is maximum, is

$\cos^{-1}({√5-1}/2)$
$\sin^{-1}({√5-1}/2)$
$π/3$
$π/5$

Validate

Solution:

OD = R sin θ
AB = 2R cos θ

\r_\OAB={\ar(\Δ\OAB)}/{\semi-\perimeter}

={1/2×\OD×\AB}/{{2\R+\AB}/2}

={1/2×\R\sin θ2\R\cos θ}/{{2\R+2\R\cos θ}/2}

\r_\OAB={\R\sin θ\cos θ}/(1+\cos θ)

{\dr_\OAB}/{dθ}={(1+\cos θ)\cos 2θ-\sin θ\cos θ(-\sin θ)}/(1+\cos θ)^2=0

\cosθ={√5-1}/2

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