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 (0, ±2) and directrices y = ±1

Verified step by step guidance

1. Identify the orientation of the hyperbola based on the vertices. Since the vertices are at (0, ±2), the hyperbola opens vertically along the y-axis.
2. Write the standard form of the hyperbola equation with vertical transverse axis centered at the origin: \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\).
3. Determine the value of \(a\) using the distance from the center to each vertex. Since vertices are at (0, ±2), \(a = 2\), so \(a^2 = 4\).
4. Use the directrix information to find the eccentricity \(e\). The directrices are given by \(y = \pm 1\), and for a hyperbola with vertical transverse axis, the directrices are at \(y = \pm \frac{a}{e}\). Set \(\frac{a}{e} = 1\) and solve for \(e\).
5. Calculate \(b^2\) using the relationship \(b^2 = a^2(e^2 - 1)\). Once \(a^2\) and \(e\) are known, substitute to find \(b^2\), 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 greater than 1, called the eccentricity (e). 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. The distance from the center to a vertex is 'a', and to a focus is 'c'. The eccentricity e = c/a relates these distances and is crucial for finding the foci and writing the hyperbola's equation.

Equation of a Hyperbola Centered at the Origin

For a hyperbola centered at the origin with vertical transverse axis, the standard form is (y²/a²) - (x²/b²) = 1. Knowing 'a' from vertices and using eccentricity to find 'c' and 'b' allows writing the equation. Directrices help determine eccentricity and complete the equation.