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Question

ABC is an isosceles triangle inscribed in a circle of radius r. If AB = AC and h is the altitude from A to BC.The triangle ABC has perimeter P=2[(2hrh2)+2hr] and A be the area of the triangle .Find limh0AP3

A
1r
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B
164r
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C
1128r
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D
12r
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Solution

The correct option is C 1128r

In ABC,AB=AC
ADBC(D is mid point of BC)
Let r = radius of circumcircle
OA = OB = OC = r
Now BD = BO2OD2
=r2(hr)2=2rhh2
BC=22rhh2
Area of ABC= 12×BC×AD
=h2rhh2
Also limh0Ap3=h2rhh28(2rhh2+2hr)3
=limh0h3/22rh8h3/2(2rh+2r)3
=limh02rh8[2rh+2r]3
=2r8(2r+2r)3=2r8.8.2r2r=1128r

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