Area under the curveHard

Question

The curve f(x) = Ax2 + Bx + C passes through the point (1, 3) and line 4x + y = 8 istangent to it at the point (2, 0). The area enclosed by y = f(x), the tangent line and the y axisis -

Options

A.4/3
B.8/3
C.16/3
D.32/3

Solution

     
Given curve is
y = f (x) =Ax2 + Bx + C        ..........(i)
It passes through (1, 3)
∴ 3 = A + B + C        ..........(ii)
point (2, 0) also lie on curve
∴ 0= 4A + 2B+ C        ..........(iii)
from (i)    = 4A + B
slope of tangent is - 4
∴ - 4 = 4A + B        ..........(iv)
∴ (ii) (iii) & (iv) we get
A = -1, B = 0, C = 4
required area = area of ᐃOAB -(-x2 + 4)dx

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