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Chapter 11: Problem 30

Find an equation for the hyperbola that satisfies the given conditions. Foci \(( \pm 6,0),\) vertices \(( \pm 2,0)\)

Short Answer

Expert verified
The hyperbola's equation is \( \frac{x^2}{4} - \frac{y^2}{32} = 1 \).

Step by step solution

01

Understanding the Hyperbola's Properties

A hyperbola is a set of points such that the difference between the distances to the two foci is constant. For a hyperbola centered at the origin, the general equation is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) for a hyperbola opening horizontally. Here, \( a \) is the distance from the center to each vertex.
02

Identify Variables from Given Information

The vertices are located at \( \pm 2, 0 \), so \( a = 2 \). The foci are located at \( \pm 6, 0 \), indicating that \( c = 6 \). In hyperbolas, \( c^2 = a^2 + b^2 \).
03

Substitute and Solve for Missing Variable

We know \( c = 6 \), \( a = 2 \), and we need to find \( b^2 \). From \( c^2 = a^2 + b^2 \), substitute the known values: \( 6^2 = 2^2 + b^2 \). Solving gives \( 36 = 4 + b^2 \). Thus, \( b^2 = 32 \).
04

Formulate the Hyperbola's Equation

Now that we have \( a^2 = 4 \) and \( b^2 = 32 \), substitute these into the standard equation for a hyperbola: \( \frac{x^2}{4} - \frac{y^2}{32} = 1 \). This is the equation for the hyperbola.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Foci of a Hyperbola
A hyperbola is closely defined by its foci, which are two distinct points that have a stable relationship with every point on the hyperbola. In essence, for any point on the hyperbola, the absolute difference in distances to the foci remains a constant. This property helps to visualize how a hyperbola stretches out between its branches.
  • In this exercise, the foci are at \(( \pm 6,0)\). This indicates that the foci lie on the x-axis, suggesting a horizontally oriented hyperbola.
  • The distance between the center of the hyperbola and each focus is denoted as \(c\).
  • For our hyperbola, \(c = 6\), connecting the position of the foci to the geometric nature of the hyperbola.
Understanding the position of the foci is vital for determining the shape and openness of the hyperbola.
Vertices of a Hyperbola
Vertices are key points on a hyperbola that indicate its widest spread along the axis it opens. They are directly related to the value 'a' in the standard hyperbola equation. Knowing the vertices allows us to measure how much the hyperbola expands from its center.
  • In our example, the vertices are positioned at \(( \pm 2, 0)\), again showing that the hyperbola stretches along the x-axis.
  • The value of \(a\), which is the distance from the center to each vertex, is \(a = 2\). This is directly plugged into the hyperbola鈥檚 standard form equation.
Understanding the placement and spacing of the vertices is crucial for determining the overall width and layout of the hyperbola's branches.
Hyperbola Properties
Understanding hyperbola properties provides us with the means to identify and distinguish a hyperbola from other conic sections. Here are a few key properties to remember:
  • A hyperbola consists of two separate branches, mirroring each other.
  • The transverse axis is the line segment that connects the vertices and contains the foci, significantly impacting the hyperbola's direction.
  • In our problem, the transverse axis lies along the x-axis, confirming a horizontal opening.
  • The conjugate axis, perpendicular to the transverse axis, affects the steepness of the branches; however, it does not intersect the branches in a real-world graph.
  • Understanding the properties such as eccentricity and asymptotes further assists in comprehending a hyperbola's behavior in detail.
These properties help us fully grasp not just the formula, but the logic and symmetry behind the hyperbola's structure.
Standard Form of Hyperbola
The standard form of a hyperbola provides a framework for understanding its shape and position. This equation relates all of the hyperbola's major components: vertices, foci, and axes.
  • The standard equation for a horizontally oriented hyperbola is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \).
  • Here, \(a\) is linked to the vertices, and \(c\) to the foci, through the equation \(c^2 = a^2 + b^2\).
  • In this exercise, substituting \(a = 2\) and \(b^2 = 32\) into the equation yields \( \frac{x^2}{4} - \frac{y^2}{32} = 1 \).
This formula is fundamental in quickly identifying the hyperbola鈥檚 features and is a template for calculating any adjustments necessary to fit different conditions.

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Most popular questions from this chapter

This exercise deals with confocal parabolas, that is, families of parabolas that have the same focus. (a) Draw graphs of the family of parabolas $$x^{2}=4 p(y+p)$$ for \(p=-2,-\frac{3}{2},-1,-\frac{1}{2}, \frac{1}{2}, 1, \frac{3}{2}, 2\) (b) Show that each parabola in this family has its focus at the origin. (c) Describe the effect on the graph of moving the vertex closer to the origin.

Find the vertices, foci, and eccentricity of the ellipse. Determine the lengths of the major and minor axes, and sketch the graph. $$ \frac{x^{2}}{16}+\frac{y^{2}}{25}=1 $$

Sunburst Window \(A\) "sunburst" window above a door- way is constructed in the shape of the top half of an ellipse, as shown in the figure. The window is 20 in. tall at its high- est point and 80 in. wide at the bottom. Find the height of the window 25 in. from the center of the base.

\begin{array}{l}{1-22 \text { a pair of parametric equations is given. }} \\\ {\text { (a) Sketch the curve represented by the parametric equations. }} \\\ {\text { (b) Find a rectangular-coordinate equation for the curve by }} \\\ {\text { eliminating the parameter. }}\end{array} $$ x=2 \sin t, \quad y=2 \cos t, \quad 0 \leq t \leq \pi $$

Spiral Path of a Dog \(\mathrm{A}\) dog is tied to a circular tree trunk of radius 1 \(\mathrm{ft}\) by a long leash. He has managed to wrap the entire leash around the tree while playing in the yard, and finds himself at the point \((1,0)\) in the figure. Seeing a squirrel, he runs around the tree counterclockwise, keeping the leash taut while chasing the intruder. (a) Show that parametric equations for the dog's path (called an involute of a circle) are \(x=\cos \theta+\theta \sin \theta \quad y=\sin \theta-\theta \cos \theta\) \([\text {Hint: Note that the leash is always tangent to the tree, }\) \(\text { so } O T \text { is perpendicular to } T D .]\) (b) Graph the path of the dog for \(0 \leq \theta \leq 4 \pi\)
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