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: $\overline{PF}- \overline {PF'}=2a$
- Foci: There are two fixed points $F$ and $F'$.
- Focal axis: It is the axis created by the straight line $FF'$ and whose length is the focal distance.
- Focal or real distance: It is the distance of the segment $\overline{FF'}=2c$.
- Secondary or imaginary axis: Axis formed by the set of equidistant points of $F$ and $F'$. It is therefore the perpendicular bisector of the segment $\overline{FF'}$.
- Center: It is the average point of the segment $\overline{FF'}$. 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 $A'$ are the points of intersection of the focal axis with the hyperbola.
- The apexes $B$ and $B'$ 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 $A'$ and with the same radius.
- Major axis: It is the axis created by the segment $\overline{AA'}$ and of length $2a$.
- Less axis: It is the axis created by the segment $\overline{BB'}$ and of length $2b$.
- Relation between semiaxes: $c^2=a^2+b^2$.
- Radioes vectors: The segments $PF$ and $PF'$, 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 $c\geq a$, dividing on both sides for $a$: $\displaystyle \frac{c}{a} \geq 1$.
The eccentricity is identified then $e \geq 1$.
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 $\displaystyle e=\frac{5}{3}$.