Problem 18 - College Algebra

Question

Use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. \(\frac{y^{2}}{25}-\frac{x^{2}}{64}-1\)

Step-by-Step Solution

Answer

The hyperbola is centered at the origin with vertices at (0, 5) and (0, -5). Its foci are located at (0, sqrt(89)) and (0, -sqrt(89)). Asymptotes equations are \(y = \frac{5}{8}x\) and \(y = -\frac{5}{8}x\).

Step 1: Recognize the form of the equation

The general equation for a hyperbola that opens upwards and downwards is given as \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\). By comparing this with the given equation, \(\frac{y^{2}}{25} - \frac{x^{2}}{64} = 1\), it can be seen that \(a^2 = 25\) and \(b^2 = 64\). This means that \(a = 5\) and \(b = 8\). Therefore, the vertices of the hyperbola are at \((0, \pm5)\).

Step 2: Locate the foci of the hyperbola

The foci of a hyperbola is given by \((0, \pm c)\) where \(c = \sqrt{a^2 + b^2}\). Hence, \(c = \sqrt{25 + 64} = \sqrt{89}\). Therefore, the foci are located at \((0, \pm \sqrt{89})\).

Step 3: Plot the hyperbola and determine the equations of asymptotes

The hyperbola can be drawn by using the vertices and asymptotes. The equations of the asymptotes for a vertical hyperbola are given by \(y = \pm\frac{a}{b}x\). On subtituting the given \(a\) and \(b\) values, we get equations of the asymptotes as \(y = \pm\frac{5}{8}x\).

Key Concepts

Vertices of a Hyperbola

Understanding the vertices of a hyperbola is essential for correctly sketching its graph. In a hyperbola, the vertices are the points where the hyperbola intersects its principal axis. This is also the axis along which the hyperbola opens.
For a vertical hyperbola of the form \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\), the vertices will be positioned at \((0, \pm a)\). To illustrate, with an equation \(\frac{y^{2}}{25} - \frac{x^{2}}{64} = 1\), we quickly determine that \(a^2 = 25\), thus \(a = 5\). As a result, the vertices are located at the coordinates \((0, 5)\) and \((0, -5)\). These points provide foundational reference for graphing the hyperbola.

Foci of a Hyperbola

The foci of a hyperbola are two fixed points located along the principal axis which, in a sense, control the shape of the hyperbola.
In the case of a hyperbola that has the equation \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\), the distance from the center to each focus is given by \(c = \sqrt{a^2 + b^2}\). Given our example equation, where \(a = 5\) and \(b = 8\), we compute \(c\) to be \(\sqrt{25 + 64} = \sqrt{89}\). Therefore, the foci of the hyperbola can be found at \((0, \sqrt{89})\) and \((0, -\sqrt{89})\). Knowing the exact location of the foci is helpful for sketching the hyperbola's characteristic `spread`, which becomes wider as the distance between the foci increases.

Equations of Asymptotes

The equations of the asymptotes are vital to understanding the behavior of a hyperbola, particularly how it extends to infinity. Asymptotes are straight lines that the hyperbola approaches but never actually touches or crosses.
For the hyperbola in vertical form \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\), the equations of asymptotes will be \(y = \pm\frac{a}{b}x\). In our hyperbola's equation, by substituting the values of \(a\) and \(b\), we acquire the asymptotes' equations as \(y = \pm\frac{5}{8}x\). These asymptotes provide a skeleton for graphing the hyperbola, and understanding them is crucial for accurately depicting how the hyperbola extends in space, ensuring that the graph flares outwards in a predictable and proportionate manner.

Conic Sections Algebra

Conic sections are the curves obtained by intersecting a plane with a cone, resulting in different shapes: circles, ellipses, parabolas, and hyperbolas. The algebraic representation of these shapes is essential for analysis and graphing.
For hyperbolas, a typical equation form is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) or the one we've been working with, \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\), where the position of \(x^2\) and \(y^2\) dictates the hyperbola's orientation—horizontal or vertical, respectively. The constants \(a\) and \(b\) determine the hyperbola's dimensions, while the variable \(c\) locates the foci. Conic sections algebra is a fundamental part of understanding the geometric properties and behaviors of shapes, allowing mathematicians and students alike to delve into the realm of analytical geometry with precision and ease.