Problem of the Week
Problem E
Points on an Ellipse

The graph of $(x+1{)}^2+(y-2{)}^2=100$ is a circle with centre $(-1,2)$ and radius $10$.

The graph of $10x^2-6xy+4x+y^2=621$ is shown below. The shape of this curve is known as an ellipse.

List all the ordered pairs $(x,y)$ of non-negative integers $x$ and $y$ that satisfy the equation $10x^2-6xy+4x+y^2=621$.

Note: When solving this problem, it might be useful to use the following idea.

By completing the square, $$x^2+y^2+2x-4y=95$$ can be rewritten as $$(x+1{)}^2+(y-2{)}^2=100$$ One solution to this equation is $(x,y)=(5,10)$.