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Intersection with Cone


Let $C$ be a double napped right circular cone whose base is $B$.

Let $\theta$ be half the opening angle of $C$.

That is, let $\theta$ be the angle between the axis of $C$ and a generatrix of $C$.

Let a plane $D$ intersect $C$.

Let $\phi$ be the inclination of $D$ to the axis of $C$.

Let $K$ be the set of points which forms the intersection of $C$ with $D$.

Then $K$ is a conic section, whose nature depends on $\phi$.


Let $\phi = \theta$.

Then $K$ is a parabola.

Focus-Directrix Property


Let $D$ be a straight line.

Let $F$ be a point.

Let $K$ be the locus of points $P$ such that the distance $p$ from $P$ to $D$ equals the distance $q$ from $P$ to $F$:

$p = q$

Then $K$ is a parabola.

Also see

  • Results about parabolas can be found here.

Historical Note

Some sources suggest that the word parabola was provided by Apollonius of Perga, who did considerable work on establishing its properties.

However, it is also believed that Menaechmus may have used the term, and that it may go back even further than that.

Linguistic Note

The word parabola is pronounced with the stress on the second syllable: par-a-bo-la.

The plural of parabola is properly parabolae, but this is considered pedantic, and the usual plural form found is parabolas.

The form parabolas is used on $\mathsf{Pr} \infty \mathsf{fWiki}$.