Find the standard form of the equation of the hyperbola satisfying the given conditions. Foci: (-8,0), (8,0); Vertices: (-3,0), (3,0)

Verified step by step guidance

1. Identify the orientation of the hyperbola: Since the foci and vertices are aligned along the x-axis, the hyperbola is horizontally oriented. The standard form of the equation for a horizontally oriented hyperbola is: $(x-h)^2/a^2\; -\; (y-k)^2/b^2\; =\; 1$, where (h, k) is the center.
2. Determine the center of the hyperbola: The center is the midpoint of the vertices. The vertices are (-3, 0) and (3, 0). Use the midpoint formula: $h\; =\; \backslash frac\{x\_1\; +\; x\_2\}\{2\}$ and $k\; =\; \backslash frac\{y\_1\; +\; y\_2\}\{2\}$. This gives the center as (0, 0).
3. Find the value of $a$: The distance from the center to each vertex is $a$. Since the vertices are at (-3, 0) and (3, 0), the distance is 3. Thus, $a\; =\; 3$, and $a^2\; =\; 9$.
4. Find the value of $c$: The distance from the center to each focus is $c$. Since the foci are at (-8, 0) and (8, 0), the distance is 8. Thus, $c\; =\; 8$, and $c^2\; =\; 64$.
5. Use the relationship $c^2\; =\; a^2\; +\; b^2$ to find $b^2$: Substitute $c^2\; =\; 64$ and $a^2\; =\; 9$ into the equation to solve for $b^2$. Once $b^2$ is determined, substitute all values into the standard form equation: $(x-0)^2/9\; -\; (y-0)^2/b^2\; =\; 1$.

Key Concepts

Hyperbola Definition

A hyperbola is a type of conic section formed by the intersection of a plane and a double cone. It consists of two separate curves called branches, which are mirror images of each other. The standard form of a hyperbola's equation can be expressed as (x-h)²/a² - (y-k)²/b² = 1 for horizontal hyperbolas, where (h, k) is the center, and 'a' and 'b' are distances related to the vertices and foci.

Foci and Vertices

In the context of hyperbolas, the foci are two fixed points located along the transverse axis, which is the line segment that connects the vertices. The vertices are the points where the hyperbola intersects its transverse axis. The distance between the center and each vertex is denoted as 'a', while the distance from the center to each focus is denoted as 'c'. The relationship between 'a', 'b', and 'c' is given by the equation c² = a² + b².

Standard Form of Hyperbola Equation

The standard form of a hyperbola's equation provides a way to represent its geometric properties algebraically. For a hyperbola centered at the origin with a horizontal transverse axis, the equation is (x²/a²) - (y²/b²) = 1. To derive this equation, one must identify the values of 'a' and 'b' based on the given foci and vertices, which allows for the complete characterization of the hyperbola's shape and orientation.