In Exercises 5–12, find the standard form of the equation of each hyperbola satisfying the given conditions. Foci: (−4, 0), (4, 0); vertices:(−3, 0), (3, 0)

Verified step by step guidance

1. Identify the center of the hyperbola. Since the vertices are at \((-3, 0)\) and \((3, 0)\), the center is the midpoint of these points, which is \((0, 0)\).
2. Determine the distance from the center to each vertex, which is the value of \(a\). The distance from \((0, 0)\) to \((3, 0)\) is 3, so \(a = 3\).
3. Identify the distance from the center to each focus, which is the value of \(c\). The distance from \((0, 0)\) to \((4, 0)\) is 4, so \(c = 4\).
4. Use the relationship \(c^2 = a^2 + b^2\) to find \(b^2\). Substitute \(c = 4\) and \(a = 3\) into the equation: \(4^2 = 3^2 + b^2\).
5. Solve for \(b^2\) to find its value, and then write the standard form of the hyperbola's equation: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).

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 from the center to each vertex is denoted as 'a', while the distance from the center to each focus is denoted as 'c'. The relationship between these distances 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 center, vertices, and foci, allowing for the calculation of 'a' and 'c', and subsequently 'b' using the relationship c² = a² + b².