Problem 19 - Calculus Early Transcendentals

Question

Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph. $$\frac{x^{2}}{144}-\frac{y^{2}}{25}=1$$

Step-by-Step Solution

Answer

Vertices: \( (12, 0), (-12, 0) \); Foci: \( (13, 0), (-13, 0) \); Asymptotes: \( y = \pm \frac{5}{12}x \).

Step 1: Identify the standard form

The given equation is \( \frac{x^{2}}{144} - \frac{y^{2}}{25} = 1 \). This is in the standard form of a hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), which indicates the hyperbola opens horizontally.

Step 2: Determine values for a, b, and c

From the equation \( \frac{x^{2}}{144} - \frac{y^{2}}{25} = 1 \), we have \( a^{2} = 144 \), so \( a = 12 \). Similarly, \( b^{2} = 25 \), so \( b = 5 \). To find \( c \), use the formula \( c^{2} = a^{2} + b^{2} \). Thus, \( c^{2} = 144 + 25 = 169 \), resulting in \( c = 13 \).

Step 3: Identify the vertices

For a horizontally oriented hyperbola with equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the vertices are located at \( (\pm a, 0) \). Therefore, the vertices are \( (12, 0) \) and \( (-12, 0) \).

Step 4: Locate the foci

The foci of the hyperbola are found at \( (\pm c, 0) \) for the horizontally oriented hyperbola. Given \( c = 13 \), the foci are at \( (13, 0) \) and \( (-13, 0) \).

Step 5: Determine the asymptotes

For a hyperbola in the form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the asymptotes are given by the lines \( y = \pm \frac{b}{a}x \). This results in the equations \( y = \pm \frac{5}{12}x \).

Step 6: Sketch the hyperbola

Plot the vertices \( (12, 0) \) and \( (-12, 0) \), the foci \( (13, 0) \) and \( (-13, 0) \), and draw the asymptotes. The hyperbola opens horizontally and approaches the asymptotes, getting farther from the center \((0,0)\) without intersecting the asymptotes.

Key Concepts

Vertices of a Hyperbola

The vertices of a hyperbola are the points where the hyperbola intersects its transverse axis. They play a crucial role in defining the size and shape of the hyperbola.
For a hyperbola of the form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), which opens horizontally, the vertices are located at the points \((\pm a, 0)\).
In our specific example, since \( a^2 = 144 \), we calculate \( a = 12 \). Therefore, the vertices are \((12, 0)\) and \((-12, 0)\).
These vertices mark the closest points between the branches of the hyperbola and help determine the width of the hyperbola.

Foci of a Hyperbola

The foci of a hyperbola are two special points located along its transverse axis. They help in defining the curvature and direction of the hyperbola.
For a horizontally opening hyperbola, as defined by the equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the foci are positioned at \((\pm c, 0)\).
To find \( c \), we use the relationship \( c^2 = a^2 + b^2 \). For our hyperbola, \( c^2 = 144 + 25 = 169 \), giving us \( c = 13 \).
Thus, the foci are located at \((13, 0)\) and \((-13, 0)\). These points help in determining the degree of opening of the hyperbola.

Asymptotes of a Hyperbola

Asymptotes are lines that a hyperbola approaches but never intersects. They determine the overall direction and spread of the hyperbola's branches.
For the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the asymptotes are given by the equations \( y = \pm \frac{b}{a}x \).
In our example, since \( a = 12 \) and \( b = 5 \), the slopes of the asymptotes are \( \pm \frac{5}{12} \), leading to the lines:

• \( y = \frac{5}{12}x \)
• \( y = -\frac{5}{12}x \)

These lines indicate the angles at which the hyperbola extends outwards, guiding the path of the graph without intersecting it.

Graphing Hyperbolas

Graphing a hyperbola involves plotting its vertices, foci, and asymptotes to understand its shape and direction.
First, locate the center of the hyperbola, usually at \((0,0)\) for the standard form equations. Then, use the vertices and foci values to mark these key points.
Next, sketch the asymptotes as dashed lines, which will help guide the curving branches of the hyperbola.
Using the vertices, draw branches of the hyperbola that extend outwards toward the asymptotes, curving away from the center.
Ensure the branches do not touch the asymptotes, as they only guide the trajectory. For a horizontal hyperbola like ours, the branches open to the left and right.

Conic Sections: Hyperbolas Explained

A hyperbola is part of the family of conic sections, which are shapes that can be formed by intersecting a plane with a cone.
Alongside circles, ellipses, and parabolas, hyperbolas offer unique geometric properties.
Hyperbolas are defined based on the differences in distances to two fixed points (the foci).
This creates two separate and symmetrical curves known as branches.
Hyperbolas have a variety of applications, including in navigation systems and in understanding electromagnetic fields. Understanding their structure and components, like vertices, foci, and asymptotes, is key to fully grasping how hyperbolas function as mathematical models.