Existence of primitive pairs with two prescribed traces over finite fields

Given F = F-pt, a field with pt elements, where p is a prime power and t >= 7 is a positive integer. Let n is an element of N and f = f(1)/f(2) is a rational function, where f(1) and f(2) are distinct irreducible polynomials with deg(f(1)) + deg(f(2)) = n <= p(t) in F[x]. We construct a sufficient condition on (p, t) which guarantees primitive pairing (is an element of, f(is an element of)) exists in F such that Tr-Fpt /F-p (is an element of) = a and Tr-Fpt /F-p (f(is an element of)) = b for any prescribed a, b is an element of F-p. Further, we demonstrate for any positive integer n <= p(t), such a pair definitely exists for large t. The scenario when n = 2 is handled separately and we verified that such a pair exists for all (p, t) except from possible 71 values of (p, t). A result for the case n = 3 is given as well.