Two curves `C_(1)equiv[f(y)]^(2//3)+[f(x)]^(1//3)=0 and C_(2)equiv[f(y)]^(2//3)+[f(x)]^(2//3)=12,` satisfying the relation `"(x-y)f(x+y)-(x+y)f(x-y)=4xy(x^(2)-y^(2))`
The area bounded by the curve `C_(1) and C_(2)` is

Options

A. (a) `2pi-sqrt(3)` sq. units
B. (b) `2pi+sqrt(3)` sq. units
C. (c) `pi+sqrt(6)` sq. units
D. (d) `2sqrt(3)-pi` sq. units

Answer

B

Explanation

Given
`(x-y)f(x+y)-(x+y)f(x-y)=4zy(x^(2)-y^(2))`
`=(x^(2)-y^(2))[(x+y)^(2)-(x-y)^(2)]`
`=(x-y)(x+y)^(3)-(x+y)(x-y)^(3)`
`rArr" "f(x+y)=(x+y)^(3)rArrf(x)=x^(3),f(y)=y^(3)`
Now equations of given curves are
`y^(2)+x=0" ...(1)"`
`x^(2)+y^(2)=12" ...(2)"`

Solving equations (1) and (2), we get `x=-3,y = pm sqrt(3)`
The area bounded by curves
`A=2[|underset(-2sqrt(3))overset(-3)intdx|+|underset(-3)overset(0)intsqrt(-x)dx|]`
`I_(1)=2overset(-3)underset(-2sqrt(3))intsqrt(12-x^(2))dx=2overset(-pi//3)underset(-pi//2)int12 cos ^(2) theta d""theta`
`=12[overset(-pi//3)underset(-pi//2)int(1+cos 2theta)d""theta]`
`=12[theta+(sin theta)/(2)]_(-pi//2)^(-pi//3)=12[-(pi)/(3)-(sqrt(3))/(4)+(pi)/(2)]`
`=12[(pi)/(6)-(sqrt(3))/(4)]=2pi-3sqrt(3).`
`I_(2)=2overset(0)underset(-3)intsqrt(-x)dx=(2[(-x)^(3//2)]_(-3)^(0))/(-3//2)=-(4)/(3)[10-3^(3//2)]=4sqrt(3).`
`A=2pi-3sqrt(3)+4sqrt(3)=2pi+sqrt(3)` sq. units.