for a>0 the line a(a^2+1)y = a - x forms a triangle with co ordinate axes then greatest area of this triangle is

we can modify the equation of the line as

as a>0 the x and y intercepts are a and 1/(a² +1)
hence the area of the triangle formed is

we need to maximize the area wrt a
so

and hence solving that we get a = 1 as a possible solution and on checking the double derivative at a=1 is -ve and hence its a maxima
so the greates area is