Problem 22 - Algebra and Trigonometry

Question

use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. $$ 4 x^{2}-25 y^{2}=100 $$

Step-by-Step Solution

Answer

The foci of the hyperbola is at (\(\sqrt{29}\), 0) and (-\(\sqrt{29}\), 0). The equations of the asymptotes are \(y = \pm \frac{2}{5}x\).

Step 1: Rewrite the equation in standard form

Rewrite the given equation \(4x^{2} - 25y^{2} = 100\) in the standard form. Divide both sides of the equation by 100 to get \(\frac{{x^{2}}{{25}} - \frac{{y^{2}}{{4}} = 1\). Thus, we get a=5 (half the length of the transverse axis) and b=2 (half the length of the conjugate axis).

Step 2: Calculate the foci

Use the equation \(c = \sqrt{a^{2} + b^{2}}\) to calculate the foci. Substituting a=5 and b=2, we get \(c = \sqrt{5^{2} + 2^{2}} = \sqrt{29}\). So, the coordinates of the foci are (\(\sqrt{29}\), 0) and (-\(\sqrt{29}\), 0).

Step 3: Find the equations of the asymptotes

The equations of the asymptotes for a hyperbola centered at origin with transverse axis on x-axis are \(y = \pm \frac{b}{a}x\). Substituting a=5 and b=2 into the equation, we find that the equations of the asymptotes are \(y = \pm \frac{2}{5}x\).

Key Concepts

Asymptotes of a Hyperbola

When studying hyperbolas, understanding the role of asymptotes is crucial. Asymptotes are imaginary lines that a hyperbola approaches as it extends towards infinity but never actually reaches. These lines are significant as they define the direction of the arms of the hyperbola and give us a clear boundary of where the graph will lie.

To find the equations of the asymptotes for a hyperbola centered at the origin, we utilize the relationships between the hyperbola's parameters. If a hyperbola is oriented horizontally, the equations of the asymptotes are given by \(y = \pm \frac{b}{a}x\), where \(a\) and \(b\) are lengths of the semi-major and semi-minor axes, respectively. In contrast, for a vertically oriented hyperbola, the asymptotes are \(x = \pm \frac{a}{b}y\).

In the provided exercise, with the hyperbola \(4x^2 - 25y^2 = 100\), we determine the asymptotes by first finding values of \(a\) and \(b\) and then plugging them into the formula for a horizontally oriented hyperbola. This yields the asymptote equations, \(y = \pm \frac{2}{5}x\). These guidelines create a 'box' or reference frame that aids in sketching the hyperbola more accurately and understanding its shape.

Foci of a Hyperbola

The foci (plural for focus) of a hyperbola are two fixed points located along the major axis that are essential for the hyperbola's definition and construction. Each point on a hyperbola is equidistant from the foci as it is from the corresponding asymptote line.

To calculate the coordinates of the foci for a hyperbola centered at the origin, we use the formula \(c = \sqrt{a^2 + b^2}\), where \(c\) represents the distance from the center to each focus, and \(a\) and \(b\) are the same values used for determining the asymptotes. For the hyperbola \(4x^2 - 25y^2 = 100\), by substituting the derived values into our formula, we find the foci to be at the coordinates (\(\sqrt{29}\), 0) and (-\(\sqrt{29}\), 0), which are located on the x-axis due to the horizontal orientation of the hyperbola. They serve as 'guiding points' that help ensure the accurate representation of the hyperbola's curvature when graphing.

Standard Form of a Hyperbola

The standard form of a hyperbola's equation is pivotal in identifying its basic features, such as orientation, centers, vertices, and asymptotes. For a hyperbola with the transverse axis aligned horizontally, the standard form is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), and if the transverse axis is vertical, then the standard form is \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\), where \(a\) is the distance from the center to vertices on the transverse axis and \(b\) is the distance to vertices on the conjugate axis.

The provided exercise requires rewriting the equation \(4x^2 - 25y^2 = 100\) into standard form. By dividing each term by 100 and rearranging, we achieve the standard form \(\frac{x^2}{25} - \frac{y^2}{4} = 1\), which clearly shows the hyperbola's orientation and the values of \(a\) and \(b\), needed to graph it. This step is essential for understanding the hyperbola's structure and is foundational in graphing the hyperbola correctly and determining other properties like the foci and asymptotes.