# 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:

$\frac{\left(x-h{\right)}^{2}}{{a}^{2}}-\frac{\left(y-k{\right)}^{2}}{{b}^{2}}=1$

Here, (h, k) represents the center of the hyperbola, 'a' is the distance from the center to the vertices along the x-axis (semi-major axis), and 'b' is the distance from the center to the vertices along the y-axis (semi-minor 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: $\frac{\left(x-h{\right)}^{2}}{{a}^{2}}-\frac{\left(y-k{\right)}^{2}}{{b}^{2}}=1$

• Vertical Hyperbola: $\frac{\left(y-k{\right)}^{2}}{{b}^{2}}-\frac{\left(x-h{\right)}^{2}}{{a}^{2}}=1$

### 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 $y=k±\frac{b}{a}\left(x-h\right)$, and for a vertical hyperbola, they are $y=k±\frac{a}{b}\left(x-h\right)$.

### 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 $\left(h±a,k\right)$, and for a vertical hyperbola, they are $\left(h,k±b\right)$.

### 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:

1. 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)

2. 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.