Hyperbola
A hyperbola is a conic section, a curve formed by intersecting a double cone with a plane. It is characterized by two distinct and symmetric branches, each extending infinitely. The general equation for a hyperbola in Cartesian coordinates is:
$\frac{(xh{)}^{2}}{{a}^{2}}\frac{(yk{)}^{2}}{{b}^{2}}=1$
Here, (h, k) represents the center of the hyperbola, 'a' is the distance from the center to the vertices along the xaxis (semimajor axis), and 'b' is the distance from the center to the vertices along the yaxis (semiminor axis).
Let's explore the key features and properties of hyperbolas.
1. Standard Forms:
The equation above is for a hyperbola centered at (h, k). There are two standard forms based on the orientation of the branches:

Horizontal Hyperbola: $\frac{(xh{)}^{2}}{{a}^{2}}\frac{(yk{)}^{2}}{{b}^{2}}=1$

Vertical Hyperbola: $\frac{(yk{)}^{2}}{{b}^{2}}\frac{(xh{)}^{2}}{{a}^{2}}=1$
2. Asymptotes:
Hyperbolas are known for their asymptotic behavior. The asymptotes are lines that the hyperbola approaches but never quite reaches. For a horizontal hyperbola, the asymptotes are given by the equations $y=k\pm \frac{b}{a}(xh)$, and for a vertical hyperbola, they are $y=k\pm \frac{a}{b}(xh)$.
3. Foci and Eccentricity:
The distance between the center (h, k) and each focus is denoted by 'c' and is related to 'a' and 'b' by the equation $c=\sqrt{{a}^{2}+{b}^{2}}$. The eccentricity ($e$) of the hyperbola is defined as $e=\frac{c}{a}$, and it determines the shape of the hyperbola.
4. Vertices:
The vertices of the hyperbola are the points where the curve intersects the transverse axis. For a horizontal hyperbola, the vertices are $(h\pm a,k)$, and for a vertical hyperbola, they are $(h,k\pm b)$.
5. Directrix:
The directrices are lines that, along with the foci, help define the shape of the hyperbola. The distance between the center and each directrix is $d=\frac{a}{e}$.
Special Cases:

Standard Form with Center at the Origin: $\frac{{x}^{2}}{{a}^{2}}\frac{{y}^{2}}{{b}^{2}}=1$ (Horizontal) $\frac{{y}^{2}}{{b}^{2}}\frac{{x}^{2}}{{a}^{2}}=1$(Vertical)

Eccentricity and Shape:
 If $e<1$, the hyperbola is an ellipse.
 If $e=1$, the hyperbola is a parabola.
 If $e>1$, the hyperbola is a hyperbola.