Let the line \(x+y=1\) meet the axes of \(x\) and \(y\) at A and B, respectively. A right angled triangle AMN is inscribed in the triangle OAB , where O is the origin and the points M and N lie on the lines \(O B\) and \(A B\), respectively. If the area of the triangle \(A M N\) is \(\frac{4}{9}\) of the area of the triangle \(O A B\) and AN : NB \(=\lambda: 1\), then the sum of all possible value(s) of is \(\lambda\) :

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.