65–68. Eccentricity-directrix approach Find an equation of the following curves, assuming the center is at the origin. Graph the curve, labeling vertices, foci, asymptotes (if they exist), and directrices.
A hyperbola with vertices (±4, 0) and directrices x = ±2

Verified step by step guidance

1. Identify the type of conic: Since the problem states it is a hyperbola with vertices at (±4, 0), the transverse axis is along the x-axis, and the center is at the origin (0,0).
2. Recall the standard form of a hyperbola centered at the origin with a horizontal transverse axis: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), where \(a\) is the distance from the center to each vertex.
3. From the vertices (±4, 0), determine \(a = 4\), so \(a^2 = 16\).
4. Use the eccentricity-directrix relationship for a hyperbola: The directrices are given by \(x = \pm \frac{a}{e}\), where \(e\) is the eccentricity. Given the directrices at \(x = \pm 2\), set \$2 = \frac{a}{e}\( and solve for \)e$.
5. Recall the relationship between \(a\), \(b\), and \(e\) for a hyperbola: \(e = \frac{c}{a}\), where \(c^2 = a^2 + b^2\). Use the value of \(e\) found to express \(b^2\) in terms of \(a^2\) and \(e\), then write the full equation of the hyperbola.

Key Concepts

Definition of a Hyperbola Using Eccentricity and Directrix

A hyperbola can be defined as the set of points where the ratio of the distance to a focus and the distance to a corresponding directrix is a constant called eccentricity (e > 1). This eccentricity-directrix definition helps derive the equation of the hyperbola when the directrices and vertices are known.

Relationship Between Vertices, Foci, and Eccentricity

Vertices are points on the hyperbola closest to the center, and foci lie along the transverse axis beyond the vertices. The distance from the center to a vertex is 'a', to a focus is 'c', and eccentricity is defined as e = c/a. Knowing vertices and directrices allows calculation of 'a', 'c', and 'e' to find the hyperbola's equation.

Standard Equation of a Hyperbola Centered at the Origin

For a hyperbola centered at the origin with a horizontal transverse axis, the standard form is (x²/a²) - (y²/b²) = 1. Here, 'a' is the distance from the center to each vertex, and 'b' relates to the conjugate axis. Using eccentricity and the relationship c² = a² + b², one can find 'b' and write the full equation.