In a circle with centre O, suppose A, P, B are three points on its circumference such that P is the mid-point of minor arc AB. Suppose when ∠AOB = θ, \frac{\area( \\Delta\AOB)}{\area( \\Delta\APB)}=\sqrt5+2 If ∠AOB is doubled to 2θ, then the ratio \frac{\area( \\Delta\AOB)}{\area( \\Delta\APB)} is

KVPY SX SB Exam 2016 Question Paper

Question : 8 of 120

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In a circle with centre O, suppose A, P, B are three points on its circumference such that P is the mid-point of minor arc AB. Suppose when ∠AOB = θ,

{\area( \Δ\AOB)}/{\area( \Δ\APB)}=√5+2

If ∠AOB is doubled to 2θ, then the ratio

{\area( \Δ\AOB)}/{\area( \Δ\APB)}

Solution:

{\Δ\AOB}/{\Δ\APB}=2+√5

{1/2 . 1 .\sin θ }/{1/2|\table 1,0,1;\cosθ/2,\sin/2,1;\cosθ,\sinθ,1|}=2+√5

⇒{\cosθ/2}/{1-\cosθ/2}=2+√5

⇒\cosθ/2={1+√5}/4

\cosθ={√5-1}/4

If θ → 2θ

{\Δ\AOB}/{\Δ\APB}= {\cosθ}/{1-\cosθ}=1/√5

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