if a and b are positive integers such that ( a^{2}-b^{4}=2009 ), then ( a+b^{2}=lambda^{2} ). The value of ( |lambda| ) is.

( quad a^{2}-b^{4}=2009 quad ; a+b^{2}=lambda^{2} )
( (a)^{2}-left(b^{2}right)^{2}=2009 )
( Rightarrowleft(a^{2}-b^{2}right)left(a+b^{2}right)=2009 )
( left(a-b^{2}right) lambda^{2}=2009:=7^{2} times 41 )
( a-b^{2} lambda ) is an integer
( begin{array}{ll}Rightarrow 41 & lambda^{2}=7^{2} Rightarrow quad | lambda |=7end{array} )