Part 2/5:
Deriving the Standard Form of the Equation
The equation of a hyperbola can be shaped through a series of geometric and algebraic manipulations. The standard form of the hyperbola centered at the origin can be expressed as:
[
\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1
]
To derive this equation, we can represent the coordinates of points on the hyperbola (x, y), and also introduce constants (a) and (b), where:
(a) is the distance from the center to the vertices on the x-axis,
(b) relates to the shape of the hyperbola.
Step-by-Step Derivation:
- Setup the Geometry: Place the hyperbola with foci at points (c, 0) and (-c, 0) on a Cartesian plane.