Of all isosceles triangles inscribed in a given circle the triangle having maximum area is .....triangle
A
Equilateral
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B
Scalene
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C
Right angled isosceles
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D
isosceles
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Solution
The correct option is C Equilateral In right △ODC, OD=acos2θ and DC=asin2θ So, AD=a+acos2θ and BC=2DC=2asin2θ Area of △ABC=12BC.AD A(θ)=a2sin2θ+a22sin4θ A′(θ)=2a2cos2θ+2a2cos4θ For maxima or minima, A′(θ)=0 cos2θ+cos4θ=0 ⇒θ=π2,π6 A′′(θ)=−6√3a2 at θ=π6 Hence, area is maximum at θ=π6 So, vertical angle BAC=2(π6)=π3 Since, ABC is isosceles, so ∠ABC=∠ACB=π3 Hence , △ABC is equilateral.