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 xaxis is (1) 3 (2) 6 (3) 9 (4) 12

Options

A. 3
B. 6
C. 9
D. 12

Answer

C

Explanation

The given parabola is `(y-2)^(2)=x-1.` So
`(dy)/(dx)=(1)/(2(y-2))`
At(2,3),
`(dy)/(dx)=(1)/(2(3-2))=(1)/(2)`
Tangent at (2,3) is
`y-3=(1)/(2)(x-2) rArr x-2y+4=0`

Therefore, the required area
`=overset(3)underset(0)int[(y-2)^(2)+1]dy-overset(3)underset(0)int(2y-4)dy`
`=|((y-2)^(3))/(3)+y|_(0)^(3)-|y^(2)-4y|_(0)^(3)`
`=(1)/(3)+3+(8)/(3)-(9-12)`
`=9` sq. units