Given x = ay² and y = ax²

=> x = a(ax² )²  

=> x = a³ x⁴

=> x - a³ x⁴ = 0

=> x(1 - a³ x³ )= 0

=> x*(1 - ax)*(1 + ax + a² x² ) = 0

=> x = 0 or 1 - ax = 0

=> x = 0 or x = 1/a

Now area = ^1/a∫ {√(x/a) - ax² }dx = 1

=> {(1/√a)*(2/3)*x^3/2 - ax³ /3 ₀}^1/a = 1 

=> (1/√a)*(2/3)*1/(a)^3/2 - a * 1/3a³ = 1

=> 2/3a² - 1/3a² = 1

=> 1/3a² = 1

=> a² = 1/3

=> a = 1/√3  (a > 0)

So, the value of a is 1/√3