Problem 23 - Precalculus: Functions and Graphs

Question

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Foci \(F(\pm 5,0)\) vertices \(V \pm 3,0)\)

Step-by-Step Solution

Answer

The hyperbola equation is \( \frac{x^2}{9} - \frac{y^2}{16} = 1 \).

Step 1: Understand the Hyperbola Features

For a hyperbola centered at the origin with a horizontal transverse axis, the standard form is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). The vertices are at \( \pm a, 0 \), and the foci are at \( \pm c, 0 \). Here, we will identify \( a \) and \( c \) based on given conditions.

Step 2: Identify Given Values

From the problem, we know the vertices \( V(\pm 3, 0) \), so \( a = 3 \). The foci are \( F(\pm 5, 0) \), so \( c = 5 \).

Step 3: Use the Relationship Between a, b, and c

In a hyperbola, the relationship between \( a \), \( b \), and \( c \) is given by \( c^2 = a^2 + b^2 \). Substitute \( a = 3 \) and \( c = 5 \) into this equation: \( 5^2 = 3^2 + b^2 \).

Step 4: Solve for b

Calculate \( b^2 \) using the equation from Step 3: \( 25 = 9 + b^2 \). Simplify to find \( b^2 = 16 \).

Step 5: Write the Hyperbola Equation

Substitute \( a^2 = 9 \) and \( b^2 = 16 \) into the standard form of the hyperbola equation: \( \frac{x^2}{9} - \frac{y^2}{16} = 1 \).

Key Concepts

Conic Sections

Conic sections are fascinating curves that result from the intersection of a plane with a double-napped cone. These sections manifest in several forms, such as circles, ellipses, parabolas, and hyperbolas. Each shape forms under specific conditions based on the angle and location of the intersecting plane.
A hyperbola, which is the focus of our discussion, forms when the intersecting plane is steeply inclined relative to the cone's axis, cutting through both nappes of the cone. Despite their appearance, hyperbolas have a symmetrical nature and exhibit unique properties. They come with two distinct branches that mirror each other, separated by a distance known as the transverse axis.
These conic sections are not only of geometric interest but also have real-world applications, including in physics and astronomy, emphasizing the importance of understanding their properties and equations.

Equation of Hyperbola

The equation of a hyperbola is key to understanding its structure and properties. When a hyperbola is centered at the origin and opens horizontally, its equation is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). This particular equation indicates that the transverse axis, which is the line segment connecting the vertices, lies along the x-axis.
In contrast, if the hyperbola opens vertically, the equation becomes \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \), with the transverse axis along the y-axis.
Here, \(a\) and \(b\) are crucial constants that determine the shape and orientation of the hyperbola. The constant \(a\) is the distance from the center to each vertex along the transverse axis, while \(b\) is related to the distance from the center to the asymptotes, invisible guide lines that aid in sketching the hyperbola.

Vertices and Foci of Hyperbola

The vertices and foci are pivotal points that define the hyperbola's shape and orientation. For a hyperbola centered at the origin, the vertices' coordinates are given by \((\pm a, 0)\) for a horizontal hyperbola or \((0, \pm a)\) for a vertical one. In our exercise, with vertices at \(V(\pm 3, 0)\), we find that \(a = 3\).
The foci are another essential part of hyperbolas. They lie along the transverse axis and are located at \((\pm c, 0)\) for a horizontal hyperbola or \((0, \pm c)\) for a vertical one. Foci are deeper in the branches compared to vertices and play a role in defining the hyperbola's eccentricity, a measure of its 'ovalness'.
In the given problem, the foci \(F(\pm 5, 0)\) tell us that \(c = 5\). These points help to predict how stretched or compressed the hyperbola appears.

Relationship Between a, b, and c

Understanding the relationship between \(a\), \(b\), and \(c\) is crucial for solving hyperbola-related exercises. The hyperbola's defining relationship is \(c^2 = a^2 + b^2\). This equation is a direct consequence of the hyperbola's geometry and derives from its distance properties.
In the context of our exercise, we already have \(a = 3\) and \(c = 5\). Substituting these into the relationship gives us \(5^2 = 3^2 + b^2\). Solving this equation, we calculate that \(b^2 = 16\), thus \(b = 4\).
This mathematical link not only helps identify \(b\), the semi-minor axis's distance, but also aids in drawing accurate graphs and understanding hyperbolas' overall properties. It ensures that each part of the hyperbola is proportionally correct, allowing for a coherent and resolved solution.