Sir there are two methods.. But one is getting right answer and other getting wrong. Please tell me where i have gone wrong. I have attached the problem and the 2 methods.. Please see to it

In your method 1 you are getting condition that (x^2+y^2)^2 is greater than 9. Speacial case of equal to is valid when your both vectors have zero angle between them. Second vector is selected arbitrarily to enforce the dot product condition to be equal to the given solution. THis gives only inequality to be valid but not to get the minimum value. Only information you get that minimum value is more than 9 but not necessarily 9. Here inequlity (>) is valid but equality (=) for minimum condition is not necessarily valid. Your answer for method 2 is the correct answer.

 

In simple language your two selected vectors are not having zero angle between them and Hence eqality condition is not the valid condition to get minimum value.