The equation for an ellipse in polar coordinates is R = p/(1+ecos(t)) where t is the angle between the periapsis and any position on the ellipse, as measured from the origin (focus).

Suppose that two ellipses in the same plane and also sharing the same focus have periapses separated by an angle D. Show that the ellipses intersect if and only if

2p₁p₂ (1 - e₁e₂cosD) is at least as large as

p₁²(1 - e₂²) + p₂²(1 - e₁²)

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Bonus problem

Suppose that the ellipses were in distinct planes, given by their normals L1 and L2. Now what is the intersection condition?

Note:

The title refers to the fact that planetary orbits are ellipses with common focus.