14. Use Structure Write the equation of the hyperbola that has its center at the origin, vertices on the y-axis 8 units apart, and asymptotes y=x. Then graph the hyperbola.
14. Use Structure Write the equation of the hyperbola that has its center at the origin, vertices on the y-axis 8 units apart, and asymptotes y=x. Then graph the hyperbola.
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
Related questions
Question
16 please
![### Practice & Problem Solving
#### UNDERSTAND
**13. Look for Relationships**
Jordan wants to graph the equation \(25x^2 - 9y^2 = 225\).
a. What kind of conic section could the equation represent? Explain.
b. Explain how to rewrite the equation in standard form. Then write the equation in standard form.
c. What are the vertices?
d. What are the asymptotes?
e. What are the foci?
f. Is the transverse axis horizontal or vertical? Explain.
g. Graph the equation.
**14. Use Structure**
Write the equation of the hyperbola that has its center at the origin, vertices on the y-axis 8 units apart, and asymptotes \(y = \pm{1.5x}\). Then graph the hyperbola.
**15. Error Analysis**
Describe and correct the error Elaine made in determining the vertices, asymptotes, and foci of a hyperbola.
\[ \text{Hyperbola: } \frac{y^2}{4} - \frac{x^2}{12} = 1 \]
Vertices: \((0 , \pm 2)\)
Asymptotes: \(y = \pm\left(\frac{1}{2}x\right)\)
Foci: \((0, -400)\) and \((0, 400)\)
**16. Higher Order Thinking**
The square of the distance from the center of a hyperbola to a focus is 78. The vertices of the hyperbola lie on the transverse axis are \((\pm 7, 0)\). Write the equation of the hyperbola.
**17. Construct Arguments**
Determine whether the equation \(\frac{18}{x^2} - \frac{52}{y^2} = 2\) represents a hyperbola. Explain.
#### PRACTICE
**18. Write an equation for the hyperbola.**
See Example 1.
Foci at \((0, \pm 15)\) and a constant difference of 30.
**19. Graph the hyperbola.**
See Example 1.
\[
\frac{y^2}{4} - \frac{x^2}{16} = 1
\]
**20. Write an equation for the hyperbola with the given information.**
See Example 2.
Asymptotes: \(y = \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5a7374f7-6e09-4d3d-b88d-a04260ba620f%2Fa74e1211-9ba6-4231-be08-001691fd6134%2Fih1cl0n_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Practice & Problem Solving
#### UNDERSTAND
**13. Look for Relationships**
Jordan wants to graph the equation \(25x^2 - 9y^2 = 225\).
a. What kind of conic section could the equation represent? Explain.
b. Explain how to rewrite the equation in standard form. Then write the equation in standard form.
c. What are the vertices?
d. What are the asymptotes?
e. What are the foci?
f. Is the transverse axis horizontal or vertical? Explain.
g. Graph the equation.
**14. Use Structure**
Write the equation of the hyperbola that has its center at the origin, vertices on the y-axis 8 units apart, and asymptotes \(y = \pm{1.5x}\). Then graph the hyperbola.
**15. Error Analysis**
Describe and correct the error Elaine made in determining the vertices, asymptotes, and foci of a hyperbola.
\[ \text{Hyperbola: } \frac{y^2}{4} - \frac{x^2}{12} = 1 \]
Vertices: \((0 , \pm 2)\)
Asymptotes: \(y = \pm\left(\frac{1}{2}x\right)\)
Foci: \((0, -400)\) and \((0, 400)\)
**16. Higher Order Thinking**
The square of the distance from the center of a hyperbola to a focus is 78. The vertices of the hyperbola lie on the transverse axis are \((\pm 7, 0)\). Write the equation of the hyperbola.
**17. Construct Arguments**
Determine whether the equation \(\frac{18}{x^2} - \frac{52}{y^2} = 2\) represents a hyperbola. Explain.
#### PRACTICE
**18. Write an equation for the hyperbola.**
See Example 1.
Foci at \((0, \pm 15)\) and a constant difference of 30.
**19. Graph the hyperbola.**
See Example 1.
\[
\frac{y^2}{4} - \frac{x^2}{16} = 1
\]
**20. Write an equation for the hyperbola with the given information.**
See Example 2.
Asymptotes: \(y = \
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