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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