In Exercises 5–12, find the standard form of the equation of each hyperbola satisfying the given conditions. Center: (4, −2); Focus: (7, −2); vertex: (6, −2)

Verified step by step guidance

1. Identify the center of the hyperbola as \((h, k) = (4, -2)\).
2. Determine the orientation of the hyperbola. Since the center, focus, and vertex all have the same y-coordinate, the hyperbola is horizontal.
3. Calculate the distance \(a\) between the center and the vertex: \(a = |6 - 4| = 2\).
4. Calculate the distance \(c\) between the center and the focus: \(c = |7 - 4| = 3\).
5. Use the relationship \(c^2 = a^2 + b^2\) to find \(b^2\), then write the standard form of the hyperbola: \(\frac{(x - 4)^2}{a^2} - \frac{(y + 2)^2}{b^2} = 1\).

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 either (x-h)²/a² - (y-k)²/b² = 1 or (y-k)²/a² - (x-h)²/b² = 1, depending on the orientation of the hyperbola.

Standard Form of a Hyperbola

The standard form of a hyperbola's equation provides a way to identify its center, vertices, and foci. The center (h, k) is the midpoint between the vertices and foci. The values 'a' and 'b' represent the distances from the center to the vertices and the distance related to the asymptotes, respectively. Understanding this form is crucial for graphing and analyzing hyperbolas.

Foci and Vertices

In a hyperbola, the foci are two fixed points located along the transverse axis, which is the line segment that connects the vertices. The distance from the center to each focus is denoted as 'c', while the distance from the center to each vertex is 'a'. The relationship between these distances is given by the equation c² = a² + b², which is essential for determining the parameters of the hyperbola.