Area under the curveHard

Question

If the tangent to the curve y = 1 - x2 at x = α, where 0 < α < 1 , meets the axes at P and Q. As α varies, the minimum value of the area of the triangle OPQ is k times the area bounded by the axes and the part of the curve for which 0 < x < 1, then k is equal to -

Options

A.
B.
C.
D.

Solution

   
Equation of tangent is  y -1+ α2 = - 2α(x - α)
so  P& Q (0, α2 + 1)
area of ᐃ OPQ (ᐃ) =
= [α3 + 2α + ]
ᐃ′ = [3α2 + 2 - ] = 0
⇒ 3α4 + 2α2 - 1 = 0 ⇒ α =
so ᐃ =
Now (1 - x2)dx
⇒    
⇒     

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