The area of the region bounded by the parabola `(y-2)^(2) = x- 1`, the tangent to the parabola at the point `(2,3)` and the x-axis is

Options

A. `2:1`
B. `3:1`
C. `3:2`
D. None of these

Answer

A

Explanation

Parabola `(y-2)^(2)=x-1`

It meets the X-axis at `B(5,0)`
Tangent to the parabola.
at `C(2,3)` is `x=2y-4`
meets the X-axis at `(-4,0)`.
Area of `DeltaABC=1/2|(2,3,1),(-4,0,1),(5,0,1)|=27/2`
Area fo the shaded region
`=int_(0)^(3)[(y-2)^(2)+1-2y+4]dy`
`=[((y-2)^(3))/3+5y-y^(2)]_(0)^(3)`
`=1/3+15-9-8/6=9`
`:.` Required ratio `=9:(27/2-9)`
`=2:1`