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The area (in sq. units) bounded by the p...

The area (in sq. units) bounded by the parabola `y=x^2-1`, the tangent at the point (2,3) to it and the y-axis is

A

`(8)/(3)`

B

`(56)/(3)`

C

`(32)/(3)`

D

`(14)/(3)`

Text Solution

Verified by Experts

The correct Answer is:
A
Given, equation of parabola is `y=x^(2)-1,` which can be rewritten as `x^(2)=y+1 or x^(2)=(y-(-1)).`
`rArr` Vertex of parabola is (0, -1) and it is open upward.
Equation of tangent at (2, 3) is given by T = 0
`rArr (y+y_(1))/(2)=x x_(1)-1, " where, " x_(1)=2`
`and y_(1)=3.`
`rArr (y+3)/(2)=2x-1`
`rArr y=4x-5`

Now, required area = area of shaded region
`=int_(0)^(2)(y("parabola")-y("tangent"))dx`
`=int_(0)^(2)[(x^(2)-1)-(4x-5)]dx `
`=int_(0)^(2)(x^(2)-4x+4)dx=int_(0)^(2)(x-2)^(2)dx `
`=|((x-2)^(3))/(3)|_(0)^(2)=((2-2)^(3))/(3)-((0-2)^(3))/(3)=(8)/(3)` sq units.
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