= Intersection of a prolate spheroid and a concentric sphere
{title2=$u^2=a^2(R^2-b^2)/(a^2-b^2)$}
For the <prolate spheroid> $u^2/a^2+(v^2+w^2)/b^2=1$ with $a>b>0$, intersect the concentric <sphere> $u^2+v^2+w^2=R^2$. When $b<R<a$, subtraction gives two <circles> in the perpendicular planes
$$
u=\pm a\sqrt{\frac{R^2-b^2}{a^2-b^2}},\qquad v^2+w^2=\frac{b^2(a^2-R^2)}{a^2-b^2}.
$$
Their centers lie on the long principal axis and the line through the centers is normal to both planes. At $R=b$ the circles merge into the equator; at $R=a$ they shrink to the two polar vertices. There is no intersection for $R<b$ or $R>a$.
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