Conic parameters: A hyperbola has eccentricity e = 2 and foci (0, ±2). Find the location of the vertices and directrices.

Verified step by step guidance

1. Identify the given parameters: eccentricity \(e = 2\) and foci at \((0, \pm 2)\). Since the foci lie on the y-axis, the hyperbola is vertical with center at the origin \((0,0)\).
2. Recall the relationship between the center, foci, and vertices for a vertical hyperbola: the foci are at \((0, \pm c)\) and the vertices are at \((0, \pm a)\), where \(c\) is the focal distance and \(a\) is the distance from the center to each vertex.
3. From the foci coordinates, determine \(c\): since the foci are at \((0, \pm 2)\), we have \(c = 2\).
4. Use the eccentricity formula for a hyperbola: \(e = \frac{c}{a}\). Given \(e = 2\) and \(c = 2\), solve for \(a\) by rearranging to \(a = \frac{c}{e}\).
5. Find the directrices using the formula for vertical hyperbolas: the directrices are horizontal lines given by \(y = \pm \frac{a}{e}\). Substitute the values of \(a\) and \(e\) to find their locations.

Key Concepts

Eccentricity of a Hyperbola

Eccentricity (e) measures how much a conic section deviates from being circular. For a hyperbola, e > 1, and it relates the distances between the foci and vertices. It is defined as e = c/a, where c is the distance from the center to a focus, and a is the distance from the center to a vertex.

Standard Form and Parameters of a Hyperbola

A hyperbola centered at the origin with vertical transverse axis has foci at (0, ±c) and vertices at (0, ±a). The relationship c² = a² + b² holds, where b relates to the conjugate axis. Knowing c and e allows calculation of a and b, which determine the shape and key points of the hyperbola.

Directrices of a Hyperbola

Directrices are fixed lines used to define a hyperbola via eccentricity. For a hyperbola with vertical transverse axis, the directrices are horizontal lines located at y = ±(a/e). They help in understanding the geometric definition and are essential for graphing and analyzing the hyperbola.