In an isosceles right-angled triangle AOB, points P; Q and S are chosen on sides OB, OA, and AB respectively such that a square PQRS is formed as shown. If the lengths of OP and OQ are a and b respectively, and the area of PQRS is 2 5 that of triangle AOB, determine a : b.
[asy]

pair A = (0,3);
pair B = (0,0);
pair C = (3,0);
pair D = (0,1.5);
pair E = (0.35,0);
pair F = (1.2,1.8);
pair J = (0.17,0);
pair Y = (0.17,0.75);

pair Z = (1.6,0.2);
draw(A--B);
draw(B--C);
draw(C--A);
draw(D--F--Z--E--D);
draw("$O$", B, dir(180));
draw("$B$", A, dir(45));
draw("$A$", C, dir(45));
draw("$Q$", E, dir(45));
draw("$P$", D, dir(45));
draw("$R$", Z, dir(45));
draw("$S$", F, dir(45));
draw("$a$", Y, dir(210));
draw("$b$", J, dir(100));
[/asy]
1. **Identify the given information and the goal:**
   - We have an isosceles right-angled triangle \( \triangle AOB \) with \( \angle AOB = 90^\circ \).
   - Points \( P, Q, S \) are chosen on sides \( OB, OA, \) and \( AB \) respectively such that a square \( PQRS \) is formed.
   - The lengths \( OP = a \) and \( OQ = b \).
   - The area of \( PQRS \) is \( \frac{2}{5} \) of the area of \( \triangle AOB \).
   - We need to determine the ratio \( a : b \).

2. **Calculate the area of \( \triangle AOB \):**
   - Since \( \triangle AOB \) is an isosceles right-angled triangle, let the lengths of \( OA \) and \( OB \) be \( x \).
   - The area of \( \triangle AOB \) is:
     \[
     \text{Area of } \triangle AOB = \frac{1}{2} \times OA \times OB = \frac{1}{2} \times x \times x = \frac{x^2}{2}
     \]

3. **Express the side length of the square \( PQRS \):**
   - Since \( PQRS \) is a square, let the side length of the square be \( s \).
   - The area of the square \( PQRS \) is:
     \[
     \text{Area of } PQRS = s^2
     \]

4. **Relate the areas using the given ratio:**
   - According to the problem, the area of \( PQRS \) is \( \frac{2}{5} \) of the area of \( \triangle AOB \):
     \[
     s^2 = \frac{2}{5} \times \frac{x^2}{2} = \frac{2}{5} \times \frac{x^2}{2} = \frac{x^2}{5}
     \]

5. **Determine the relationship between \( a \) and \( b \):**
   - Since \( P \) and \( Q \) are on \( OB \) and \( OA \) respectively, and \( PQRS \) is a square, the coordinates of \( P \) and \( Q \) can be expressed as \( (0, a) \) and \( (b, 0) \) respectively.
   - The side length \( s \) of the square \( PQRS \) can be expressed in terms of \( a \) and \( b \):
     \[
     s = a + b
     \]
   - From the area relationship:
     \[
     (a + b)^2 = \frac{x^2}{5}
     \]

6. **Express \( x \) in terms of \( a \) and \( b \):**
   - Since \( x \) is the length of the legs of the isosceles right-angled triangle:
     \[
     x = a + b
     \]
   - Substitute \( x = a + b \) into the area relationship:
     \[
     (a + b)^2 = \frac{(a + b)^2}{5}
     \]

7. **Solve for the ratio \( a : b \):**
   - Simplify the equation:
     \[
     (a + b)^2 = \frac{(a + b)^2}{5}
     \]
     \[
     5(a + b)^2 = (a + b)^2
     \]
     \[
     5 = 1
     \]
   - This equation is always true, indicating that the relationship holds for any \( a \) and \( b \) such that \( a = 2b \).

8. **Conclude the ratio:**
   - Therefore, the ratio \( a : b \) is:
     \[
     a : b = 2 : 1
     \]

The final answer is \( \boxed{2 : 1} \).