Study of the hyperbola

A hyperbola is the curve formed by the set of points of the plane, for which the difference of distances to two fixed points, the foci, is constant: $\overline{PF}-\overline{P{F}^{\prime }}=2a$

• Foci: There are two fixed points $F$ and $F^{\prime }$.
• Focal axis: It is the axis created by the straight line $F{F}^{\prime }$ and whose length is the focal distance.
• Focal or real distance: It is the distance of the segment $\overline{F{F}^{\prime }}=2c$.
• Secondary or imaginary axis: Axis formed by the set of equidistant points of $F$ and $F^{\prime }$. It is therefore the perpendicular bisector of the segment $\overline{F{F}^{\prime }}$.
• Center: It is the average point of the segment $\overline{F{F}^{\prime }}$. Also, it is the point where the focal axis and the secondary axis intersect.
• Symmetry axes: Both the focal axis and the secondary axis are symmetry axes.
• Apexes: The apexes $A$ and $A^{\prime }$ are the points of intersection of the focal axis with the hyperbola.
• The apexes $B$ and $B^{\prime }$ are obtained with the intersections of the secondary axis with the center circle $A$ and of radius $c$.
• For symmetry they are found with the center circle $A^{\prime }$ and with the same radius.
• Major axis: It is the axis created by the segment $\overline{A{A}^{\prime }}$ and of length $2a$.
• Less axis: It is the axis created by the segment $\overline{B{B}^{\prime }}$ and of length $2b$.
• Relation between semiaxes: $c^2=a^2+b^2$.
• Radioes vectors: The segments $PF$ and $P{F}^{\prime }$, that join the foci with a point of the hyperbola.
• Asymptotes: A hyperbola has two asymptotes of respective equations ${\displaystyle y=\frac{b}{a}x}$ and ${\displaystyle y=-\frac{b}{a}x}$.

Eccentricity

The eccentricity gives us information about the gap in the branches of the hyperbola. $${\displaystyle e=\frac{c}{a}}$$ As $c\ge a$, dividing on both sides for $a$: ${\displaystyle \frac{c}{a}\ge 1}$.

The eccentricity is identified then $e\ge 1$.

In the extreme case $e=1$ the branches are horizontal. As the eccentricity increases more and more the branches of the hyperbola are more vertical as one sees with ${\displaystyle e=\frac{5}{4},e=\sqrt{2}}$ (equilateral hyperbola) and ${\displaystyle e=\frac{5}{3}}$.