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Chapter 11: Problem 8

What is the equation of the standard hyperbola with vertices at \((0, \pm a)\) and foci at \((0, \pm c) ?\)

Short Answer

Expert verified
Answer: The equation of the standard hyperbola is given by \(\frac{y^2}{a^2} - \frac{x^2}{c^2 - a^2} = 1\).

Step by step solution

01

Understand the hyperbola's orientation

The given vertices and foci are on the vertical axis, which means that the hyperbola is vertically oriented. For a vertically oriented hyperbola, the equation is of the form \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\).
02

Find the value of a

We are given the coordinates of vertices as \((0, \pm a)\). So, the value of \(a\) is already given and does not need to be found.
03

Find the value of c

We are given the coordinates of foci as \((0, \pm c)\). So, the value of \(c\) is also given and does not need to be found.
04

Determine the relationship between a, b, and c

For a hyperbola, the relationship between the semi-major axis (\(a\)), semi-minor axis (\(b\)), and the distance between the foci and center (\(c\)) is: \(c^2 = a^2 + b^2\). We have the values of \(a\) and \(c\), so we can use this equation to find the value of \(b^2\).
05

Solve for b^2

Using the relationship \(c^2 = a^2 + b^2\), we substitute the given values of \(a\) and \(c\) and solve for \(b^2\). We get: \(b^2 = c^2 - a^2\)
06

Write the equation of the hyperbola

Finally, we substitute the values of \(a^2\) and \(b^2\) in the equation \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\), which gives us the equation of the standard hyperbola: $$\frac{y^2}{a^2} - \frac{x^2}{c^2 - a^2} = 1$$

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Most popular questions from this chapter

The butterfly curve of Example 8 may be enhanced by adding a term: $$r=e^{\sin \theta}-2 \cos 4 \theta+\sin ^{5}(\theta / 12), \quad \text { for } 0 \leq \theta \leq 24 \pi$$ a. Graph the curve. b. Explain why the new term produces the observed effect.

Sketch the graph of the following ellipses. Plot and label the coordinates of the vertices and foci, and find the lengths of the major and minor axes. Use a graphing utility to check your work. $$x^{2}+\frac{y^{2}}{9}=1$$

Graph the following equations. Then use arrows and labeled points to indicate how the curve is generated as \(\theta\) increases from 0 to \(2 \pi\). $$r=\frac{1}{1+2 \cos \theta}$$

Give the property that defines all ellipses.

Sketch the graph of the following ellipses. Plot and label the coordinates of the vertices and foci, and find the lengths of the major and minor axes. Use a graphing utility to check your work. $$\frac{x^{2}}{5}+\frac{y^{2}}{7}=1$$
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