Finding integer solutions to a quadratic equation in 2 variables

Question

We have an equation $x^2+4y^2-2xy-2x-4y-8=0$. Find all integer pairs $(x,y)$ satisfying this equation. I did some research on my own, and found that the above equation describes an ellipse. But I'm not sure how it helps.

Is there any systematic way to solve this?

Answer 1

You can write your equation as $$(x-y-1)^2+3(y-1)^2=12.$$ That means $|y-1|\le\sqrt{12/3}=2$, it's rather straightforward to check those few values.

Answer 2

Hint: as an equation in $x$ the reduced discriminant of $x^2-2x(y+1)+4y^2-4y-8=0$ is:

$$ \frac{1}{4} \Delta = (y+1)^2-(4y^2-4y-8)=-3y^2+6y+9=-3(y+1)(y-3) $$

For integer solutions, the reduced discriminant must be a perfect square.