Problem 31 - Algebra and Trigonometry

Question

Find an equation for the hyperbola that satisfies the given conditions. Vertices \(( \pm 1,0),\) asymptotes \(y=\pm 5 x\)

Step-by-Step Solution

Answer

The equation of the hyperbola is \(x^2 - \frac{y^2}{25} = 1\).

Step 1: Determine the Orientation of the Hyperbola

Since the vertices of the hyperbola are at \(( \pm 1,0)\), the hyperbola is centered at the origin \((0,0)\). The vertices lie on the x-axis, indicating that it is a horizontally oriented hyperbola.

Step 2: Identify the Standard Form of the Equation

The standard form of a horizontally oriented hyperbola is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), where \(a\) is the distance from the center to each vertex.

Step 3: Find the Value of \(a\)

Since the vertices are at \(( \pm 1,0)\), the distance from the center \((0,0)\) to each vertex \(( \pm 1,0)\) is \(a = 1\). {Therefore, \(a^2 = 1^2 = 1\).

Step 4: Determine the Relationship Between \(a\), \(b\), and the Slope of the Asymptotes

The slopes of the asymptotes for a horizontally oriented hyperbola are given by \(\pm \frac{b}{a}\). In this case, the asymptotes are \(y = \pm 5x\), hence \(\frac{b}{a} = 5\).

Step 5: Solve for \(b\)

Since \(a = 1\), we substitute \(a\) into \(\frac{b}{a} = 5\) to get \(\frac{b}{1} = 5\), which simplifies to \(b = 5\).Consequently, \(b^2 = 5^2 = 25\).

Step 6: Write the Equation of the Hyperbola

Substitute \(a^2 = 1\) and \(b^2 = 25\) into the standard form equation \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), resulting in:\[\frac{x^2}{1} - \frac{y^2}{25} = 1\] which simplifies to:\[x^2 - \frac{y^2}{25} = 1\]

Key Concepts

Understanding Horizontal Hyperbola

A horizontal hyperbola is a type of hyperbola that is stretched along the x-axis. To identify if a hyperbola is horizontal, you can look at the position of its vertices. If the vertices are aligned along the x-axis, it indicates a horizontal orientation. This means the hyperbola opens left and right. In the given problem, the vertices are located at \((\pm 1, 0)\), which clearly places them along the x-axis. As such, this confirms our hyperbola is indeed horizontal. Being horizontal affects how we write its standard equation, and how we think about its other features.

Vertices of Hyperbola Explained

The vertices of a hyperbola are crucial because they determine important aspects of its shape. Think of vertices as the "tips" of the hyperbola. They are the points close to the center where the hyperbola curves inward.
For this exercise, the vertices are given at \((\pm 1, 0)\). These points are precisely one unit away from the center at the origin, \((0, 0)\).
The distance from the center to either vertex is denoted by the letter \(a\). In this case, \(a = 1\). Knowing \(a\) helps in forming the standard equation and also links to the slope of the asymptotes.

Deciphering Asymptotes of Hyperbola

Asymptotes play a vital role in sketching and understanding the behavior of a hyperbola. These are diagonal lines that the arms of the hyperbola approach but never intersect. They guide the hyperbola's shape and behaviour.
In our example, the asymptotes given are \(y = \pm 5x\). This reflects the slope \(\pm 5\). For horizontal hyperbolas, the slope of the asymptotes is expressed as \(\pm \frac{b}{a}\). Given \(a = 1\), we set the slope equal to the value we have, resulting in \( \frac{b}{1} = 5\). Hence, \(b = 5\). This simplifies the relationship between \(a\), \(b\), and the slope.

The Standard Form of Hyperbola

The standard form is a specific equation structure used for representing a hyperbola mathematically. For horizontal hyperbolas, the standard form is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). This form is crucial because it reveals important information about the hyperbola, including its shape and orientation.
Substituting the known values from our exercise: with \(a = 1\) (from the vertices), we have \(a^2 = 1\), and \(b = 5\) (from the asymptotes), hence \(b^2 = 25\).
The equation of the hyperbola becomes \(x^2 - \frac{y^2}{25} = 1\). This is the efficient representation of the hyperbola following the given conditions.