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Question

# Consider the collection of all curve of the form y=a−bx2 that pass through the point (2,1), where a and b are positive constant. Determine the value of a and b that will minimise, the area of the region bounded by y=a−bx2 and x-axis. Also find the minimum area

A
b=1/8,Aminimum=43sq.units
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B
b=1/4,Aminimum=43sq.units
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C
b=1/4,Aminimum=23sq.units
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D
b=1/8,Aminimum=23sq.units
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Solution

## The correct option is C b=1/8,Aminimum=4√3sq.unitsCurve y=a−bx2 passes through the point (2,1)∴a−4b=1A=2∫√a/b0(a−bx2)dx=2[ax−bx33]√a/b0=43a3/2√b=43(1+4b)3/2√b⇒A′=23√1+4b(8b−1)b3/2⇒A′=0⇒b=18⇒A=4√3sq.units

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