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. 3
B. 6
C. 9
D. 12

Answer

c

Explanation

The equation of the parabola is `y^(2)-4y-x+5=0`
The equation of tangent at (2,3) is
`3y-2(y+3)-((x+2))/(2)+5=0or, 2y-x-4=0`

`therefore` Required area A is given by
`A=underset(0)overset(3)(int)(x_(2)-x_(1))dx`
`impliesA=underset(0)overset(3)(int)[{(y-2)^(2)+1}-{2y-4}]dy`
`impliesA=underset(0)overset(2)(int)(y^(2)-6y+9)dy=underset(0)overset(3)(int)(3-y)^(2)dy=-[((3-y)^(3))/(3)]_(0)^(3)=9`