Positive ordered pair of $(10x)^2+(5y)^2<1,000,000$ How can I find the number of ordered pair $(x,y)\in\mathbb{Z}_{>0}$ where $(10x)^2+(5y)^2<1,000,000$?
Thanks

To get an approximate answer, note that the ellipse with axes [-100,100] in $x$ and $[-200,200]$ in $y$ has area $20000\pi\approx 62864$.  This is the approximate number of lattice points in the ellipse.  $600$ are on the axes, and a quarter of what is left is in the first quadrant, so there are approximately $\frac 14 \cdot 62264=15566$ solutions
