In this paper, we examine the Pell-type equation \(x^2-axy+ay^2=1\) , which is a quadratic Diophantine equation with mixed terms. Using the continued fraction expansion of an associated quadratic irrational, we derive the first positive integer solution. All positive integer solutions are then expressed explicitly in terms of generalized Fibonacci, Lucas, Pell, and Pell–Lucas sequences. Recurrence relations and closed-form expressions for the solution pairs are obtained. This approach establishes a unified connection between mixed quadratic forms, continued fractions, and linear recurrence sequences, and is applicable to other Pell-type equations. The work provides an explicit treatment of a mixed-term Pell-type equation that extends beyond standard Pell theory.
