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 \((x,y)=(5,10)\).