Question

how many integer pairs (x y) satisfy x^2+4y^2-2xy-2x-4y-8=0

Answer 1

In such questions, try to make perfect squares of linear equations. By the way, this equation represents an ellipse (you can check that the delta or determinant value is actually non zero and eccentricity is between 0 and 1). And, all the integral solutions of this equation lie on horizontal and vertical lines (symmetry of ellipse).

Hope, this helps...