Did you know, that when you take slices through a cone at different angles, you get cross-sections shaped like circles (slicing parallel to the base), ellipses (slicing at a slight angle to the base), parabolas (slicing parallel to the edge of the cone) and hyperbolas (slicing at a steep angle to the base)? We call all of these curves conic sections because they're obtained by taking different cross-sections of a cone.

We can also define these curves as the sets of points satisfying a condition that involves a straight line (or two) called the directrix (or directrices) and a point (or two) called the focus (or foci). Ellipses and Hyperbolas have two foci and two directrices. Parabolas have only one.

The points on a conic section move so that the ratio between the

• Distance from the point to one focus and the
• Perpendicular distance from the point to one directrix

• For ellipses, the ratio between these distances is always less than one.
• For parabolas, the ratio is always equal to one. The two distances are always equal.
• For hyperbolas, the ratio is always greater than one.

Different types of conic sections have different eccentricities:

• All circles have \(e=0\)
• The eccentricity of ellipses varies: \(0 .
• All parabolas have \(e = 1\).
• Hyperbolas all have \(e > 1\).
• Being as uncurved as you can get, straight lines have infinite eccentricity.

In a parabola, the length of the latus rectum is equal to four times the focal length (the distance from the focus to the vertex).

An ellipse has two latus recta (the plural of latus rectum) because it has two foci and two directrices.

The length of the latus rectum is equal to \(\dfrac{2b^2}{a}\), where \(a\) is half the length of the major axis and \(b\) is half the length of the minor axis.