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Chapter 10: Problem 47

Find an equation of the following hyperbolas, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, and asymptotes. Use a graphing utility to check your work. A hyperbola with vertices (卤2,0) and asymptotes \(y=\pm 3 x / 2\)

Short Answer

Expert verified
Answer: The equation of the hyperbola is \(\frac{x^2}{4} - \frac{y^2}{9} = 1\), the foci are at the points (卤\(\sqrt{13}\), 0), and the asymptotes are \(y = \pm \frac{3x}{2}\).

Step by step solution

01

Identify the vertices

The vertices of the hyperbola are given as (卤2, 0). Since the vertices are located at (卤a, 0), we know that a = 2.
02

Find the slope of the asymptotes

The asymptotes of the hyperbola are given as \(y = \pm 3 x / 2\). Since the slopes of the asymptotes are \(卤\frac{b}{a}\), we can find the value of b by setting \(\frac{b}{a} = \frac{3}{2}\).
03

Solve for b

We already know the value of a, which is 2. We can now solve for b by plugging the value of a and the slope of the asymptotes into the equation \(\frac{b}{a} = \frac{3}{2}\): $$ b = 3 \cdot \frac{2}{2} = 3 $$
04

Find the equation of the hyperbola

Now that we have the values of a and b, we can plug them into the standard equation of a hyperbola to find the equation: $$ \frac{x^2}{2^2} - \frac{y^2}{3^2} = 1 $$ So the equation of the hyperbola is: $$ \frac{x^2}{4} - \frac{y^2}{9} = 1 $$
05

Find the foci

To find the foci of the hyperbola, we'll use the relationship \(c^2 = a^2 + b^2\). Since we already have the values of a and b: $$ c^2 = 2^2 + 3^2 $$ $$ c^2 = 4 + 9 $$ $$ c^2 = 13 $$ $$ c = \sqrt{13} $$ So the foci are at the points (卤\(\sqrt{13}\), 0).
06

Sketch the graph

Now that we have all the information we need, we can sketch the graph of the hyperbola. Label the vertices (卤2, 0), the center (0, 0), the foci (卤\(\sqrt{13}\), 0), and draw the asymptotes \(y = \pm 3x / 2\). The final graph should look like this: [Graph of the hyperbola showing the vertices, foci, and asymptotes]
07

Check your work with a graphing utility

To confirm the correctness of the equation and the graph, use a graphing utility to plot the equation of the hyperbola \(\frac{x^2}{4} - \frac{y^2}{9} = 1\). Make sure the graph shows the correct vertices, foci, and asymptotes as calculated above.

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Key Concepts

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

Vertices and Foci
Understanding the vertices and foci of a hyperbola is crucial as they define the shape and orientation of the hyperbola. In the given problem, the vertices are at (卤2, 0). This tells us that the hyperbola opens horizontally along the x-axis. Here, "卤2" represents the distance from the center of the hyperbola, located at the origin (0,0), to each vertex.

To find the foci, we use the relationship involving the semi-major axis (a) and the semi-minor axis (b) of the hyperbola given by the equation: \[ c^2 = a^2 + b^2 \]where c is the distance from the center to each focus. In our solution:
  • We have a = 2, since the vertices are at 卤2.
  • The value of b, calculated from the slopes of the asymptotes, is 3.
Plugging these into the equation, we solve for c:\[ c^2 = 2^2 + 3^2 = 13 \]\[ c = \sqrt{13} \]Thus, the foci are located at (卤\(\sqrt{13}\), 0), further clarifying the hyperbola's structure.
Asymptotes in Hyperbolas
Asymptotes are the lines that the hyperbola approaches but never meets as it extends toward infinity. They are important for sketching the hyperbola accurately. In the exercise, the asymptotes are given by the equations:\[ y = \pm \frac{3}{2}x \]These lines form diagonals that cross through the center of the hyperbola.

The slopes of these asymptotes, \(\pm \frac{3}{2}\), give us the ratio \(\frac{b}{a}\), where b is the semi-minor axis and a is the semi-major axis. Since we already calculated that a = 2, we can confirm b as follows:
  • \(\frac{b}{2} = \frac{3}{2}\)
  • Solving gives \(b = 3\)
These calculated slopes match, verifying that these are indeed the correct asymptotes for this hyperbola. The asymptotes help guide the graph's shape by forming a "frame" that the curves approach.
Graphing Hyperbolas
Graphing a hyperbola involves plotting various components: the asymptotes, vertices, and foci. Together, these guide the accurate representation of the hyperbola.
  • Step 1: Begin by marking the center of the hyperbola, which is at the origin (0,0).
  • Step 2: Plot the vertices found at (卤2, 0). These are the points that define the "turning points" of the hyperbola.
  • Step 3: Next, draw the asymptotes using the equations \(y = \pm \frac{3}{2}x\) through the center. These will appear as straight diagonal lines crossing at the origin.
  • Step 4: Indicate the foci at (卤\(\sqrt{13}\), 0). These points lie further out along the x-axis and slightly influence the curve's appearance.
The hyperbola itself consists of two separate curves that approach but never touch the asymptotes as they extend. To complete the graph, draw smooth curves starting from each vertex, hugging the asymptotes and expanding outward. Checking the graph using a graphing utility ensures all features like vertices, foci, and asymptotes are accurately represented.

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Most popular questions from this chapter

A general hypocycloid is described by the equations $$\begin{aligned}&x=(a-b) \cos t+b \cos \left(\frac{(a-b) t}{b}\right)\\\&y=(a-b) \sin t-b \sin \left(\frac{(a-b) t}{b}\right)\end{aligned}$$ Use a graphing utility to explore the dependence of the curve on the parameters \(a\) and \(b\)

Use a graphing utility to graph the parabolas \(y^{2}=4 p x,\) for \(p=-5,-2,-1,1,2,\) and 5 on the same set of axes. Explain how the shapes of the curves vary as \(p\) changes.

Suppose that the ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1\) is revolved about the \(x\) -axis. What is the volume of the solid enclosed by the ellipsoid that is generated? Is the volume different if the same ellipse is revolved about the \(y\) -axis?

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The hyperbola \(x^{2} / 4-y^{2} / 9=1\) has no \(y\) -intercepts. b. On every ellipse, there are exactly two points at which the curve has slope \(s,\) where \(s\) is any real number. c. Given the directrices and foci of a standard hyperbola, it is possible to find its vertices, eccentricity, and asymptotes. d. The point on a parabola closest to the focus is the vertex.

The Lam茅 curve described by \(\left|\frac{x}{a}\right|^{n}+\left|\frac{y}{b}\right|^{n}=1,\) where \(a, b,\) and \(n\) are positive real numbers, is a generalization of an ellipse. a. Express this equation in parametric form (four pairs of equations are needed). b. Graph the curve for \(a=4\) and \(b=2,\) for various values of \(n\) c. Describe how the curves change as \(n\) increases.
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