Let S is the region of points which satisfies `y^(2)lt16x,x lt4 and (xy(x^(2)-3x+2))/(x^(2)-7x+12)gt0`. Its area is

Options

A. `(8)/(3)`
B. `(64)/(3)`
C. `(32)/(3)`
D. none of these

Answer

B

Explanation

`(xy(x^(2)-3x+2))/(x^(2)-7x+12)gt0`
`rArr" "(xy(x-1)(x-2))/((x-3)(x-4))gt0`
`rArr" "{{:(ygt0",","if",(x(x-1)(x-2))/((x-3)(x-4))gt0),(ylt0",","if",(x(x-1)(x-2))/((x-3)(x-4))lt0):}`
`rArr" "{{:(ygt0",","if",x in(0,1)uu(2,3)uu(4,oo)),(ylt0",","if",x in(-oo,0)uu(1,2)uu(3,4)):}`
`y^(2)lt16x" is interior of the parabola "y^(2)=16x`
Region is as shown in the following figure:

From the figure, required area
= Half of the area of the region bounded by
`y^(2)=16x and x=4`