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:

(xh)2a2(yk)2b2=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: (xh)2a2(yk)2b2=1

  • Vertical Hyperbola: (yk)2b2(xh)2a2=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±ba(xh), and for a vertical hyperbola, they are y=k±ab(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=a2+b2. The eccentricity (e) of the hyperbola is defined as e=ca, 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±a,k), and for a vertical hyperbola, they are (h,k±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=ae.

Special Cases:

  1. Standard Form with Center at the Origin: x2a2y2b2=1 (Horizontal)    y2b2x2a2=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.