Hyperbolas and Eccentricity

Exercises 25–28 give the eccentricities and the vertices or foci of hyperbolas centered at the origin of the xy-plane. In each case, find the hyperbola’s standard-form equation in Cartesian coordinates.

Eccentricity: 1.25
Foci: (0, ±5)

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Identify the orientation of the hyperbola based on the foci coordinates. Since the foci are at (0, ±5), the hyperbola opens vertically along the y-axis.

Recall the standard form of a hyperbola centered at the origin with vertical transverse axis: \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = -1\) or equivalently \(\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1\). For vertical hyperbolas, the form is \(\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1\).

Use the relationship between the eccentricity \(e\), the distance to the foci \(c\), and the distance to the vertices \(a\): \(e = \frac{c}{a}\). Given \(e = 1.25\) and \(c = 5\) (from the foci coordinates), solve for \(a\) using \(a = \frac{c}{e}\).

Calculate \(b^{2}\) using the relationship for hyperbolas: \(c^{2} = a^{2} + b^{2}\). Rearrange to find \(b^{2} = c^{2} - a^{2}\).

Write the standard form equation of the hyperbola using the values of \(a^{2}\) and \(b^{2}\) found, in the form \(\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1\).

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Definition of a Hyperbola

A hyperbola is a set of points in a plane where the difference of the distances to two fixed points called foci is constant. It has two branches and can be oriented horizontally or vertically. The standard form of its equation depends on the orientation and the location of its center.

Eccentricity of a Hyperbola

Eccentricity (e) measures the 'flatness' of a hyperbola and is defined as the ratio of the distance from the center to a focus (c) over the distance from the center to a vertex (a), i.e., e = c/a. For hyperbolas, e > 1, and knowing e helps relate the distances c and a to write the equation.

Standard Form 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. Here, a is the distance from the center to each vertex, c is the distance to each focus, and b is found using the relationship c² = a² + b². This form allows writing the equation once a, b, and c are known.

Hyperbolas and EccentricityExercises 25–28 give the eccentricities and the vertices or foci of hyperbolas centered at the origin of the xy-plane. In each case, find the hyperbola’s standard-form equation in Cartesian coordinates.Eccentricity: 1.25Foci: (0, ±5)