Parabola

Synthetically, a parabola is the locus of points in a plane that are equidistant from a given line (the directrix) and a given point (the focus). Alternatively, a parabola is a conic section obtained as the intersection of a right circular cone with a plane parallel to a generator of the cone.

Let $d$ be a line and $F$ a point. In the special case when $F$ is a point of $D$, the "parabola" with directrix $D$ and focus $F$ is the line through $F$ that is perpendicular to $D$. In the language of conic sections, this corresponds to the case when the plane contains a generator of the cone.

To avoid this degenerate case, assume that $F$ does not lie in $D$, let $\Pi$ be the unique plane containing $F$ and $D$ and let $P$ be the parabola with focus $F$ and directrix $d$. The line Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s} through Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} and perpendicular to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D} is called the axis of the the parabola Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} and is the unique line of symmetry of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} . The unique point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s} that is equidistant from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D} lies on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} and is known as the vertex of the parabola, and the distance $FV$ (half the distance from $F$ to $D$) is called the focal distance of the parabola.

Now let $P'$ be any other parabola in $\Pi$ be with the same focal distance as $P$. Let $F'$ be its focus and $D'$ its directrix. Then there is a unique, orientation-preserving rigid motion of $\Pi$ taking $F$ to $F'$ and $D$ to $D'$ and therefore, the parabola $P$ to the parabola $P'$. In other words, any two parabolas with the same focal distance are congruent.