If `' a^(prime)(a >0)` is the value of parameter for each of which the area of the figure bounded by the straight line `y=(a^2-a x)/(1+a^4)` and the parabola `y=(x^2+2a x+3a^2)/(1+a^4)` is the greatest, then the value of `a^4` is___

Answer

3

Explanation

`y=(a^(2)-ax)/(1+a^(4))" (1)"`
`=(x^(2)+2ax+3a^(2))/(1+a^(4))c" (2)"`
Point of intersection of (1) and (2)
`(a^(2)-ax)/(1+a^(4))=(x^(2)+2ax+3a^(2))/(1+a^(4))`
`(x+a)(x+2a)=0`
`x=-a,-2a`
`"Req. area "=overset(-a)underset(-2a)int[((a^(2)-ax)/(1+a^(4)))-((x^(2)+2ax+3a^(2))/(1+a^(4)))]`
`therefore" "f(a)=(a^(3))/(6(1+a^(4)))`
If f(a) is max, then f'(a)=0
`rArr" "3+3a^(4)-4a^(4)=0`
`"or "a^(4)=3`