Given $ a *b =a^2+b^2$
Let e be an identity element of * operation
By definition of identity element we have a*e=e*a = a
but by definition of * operation $ a *e =a^2+e^2$ =>$ a=a^2+e^2$ =>$ e^2= a-a^2$
=> e = square root of$ a=a^2$, but square root of $a=a^2$ does not always belong to N for all a in N
Therefore the operation * defined by $ a *b =a^2+b^2$ has no identity.