Definitions of ellipse, hyperbola and parabola

Now, let us revive the definitions of ellipse, hyperbola and parabola.

Ellipse is a set of all points P in a plane with the following property: for each such point P, the sum of its distances from the two arbitrary given points F₁ and F₂ in that plane (points F₁ and F₂ are called foci) is constant. Mathematically speaking the following relation must be valid:

|F₁P| + |PF₂| = 2a > |F₁F₂|,

where 2a is a constant greater than the distance of foci.

Hyperbola is a set of all points P in a plane with the following property: for each such point P, the absolute value of the difference of its distances from the two arbitrary given points F₁ and F₂ in that plane (points F₁ and F₂ are again called foci) is constant.

Mathematically speaking, the following relation must be valid:

||F₁P| – |PF₂|| = 2a < |F₁F₂|,

where 2a is a constant smaller than the distance of foci. Hyperbola consists of two branches. If a point P is located on one of the branches, relation |F₁P| – |PF₂| = 2a is valid. If, on the other hand, point P is located on the other branch, relation |F₂P| – |PF₁| = 2a is valid.

Parabola is a set of all points P in a plane with the following property: for each such point P, its distance from an arbitrary given line p (called directrix), lying in that plane, must be the same as its distance from an arbitrary given point F (called focus) in that plane. Point P does not lie on the line p. Mathematically speaking, the following relation must be valid: |FP| = |Pp|.