Find an equation for the hyperbola described. Graph the equation. Center at (8,-1); focus at (10,- 1); vertex at (9, -1) Write an equation for the hyperbola. 0-0-1 (Type exact answers for each term, using fractions as needed.)
Find an equation for the hyperbola described. Graph the equation. Center at (8,-1); focus at (10,- 1); vertex at (9, -1) Write an equation for the hyperbola. 0-0-1 (Type exact answers for each term, using fractions as needed.)
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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### Hyperbola Equation and Graphing
#### Problem Statement:
Find an equation for the hyperbola described. Graph the equation.
Given:
- Center at \( (8, -1) \)
- Focus at \( (10, -1) \)
- Vertex at \( (9, -1) \)
#### Task:
Write an equation for the hyperbola.
Equation Format (exact answers for each term, using fractions as needed):
\[ \frac{\Box}{\Box} - \frac{\Box}{\Box} = 1 \]
---
**Instructions:**
1. **Identify the Elements:**
- Center: \( (h, k) = (8, -1) \)
- Vertices: The distance from the center to a vertex (along the transverse axis) is the distance from \( (8, -1) \) to \( (9, -1) \), which is \( a \).
- Foci: The distance from the center to a focus (focal distance) is the distance from \( (8, -1) \) to \( (10, -1) \), which is \( c \).
2. **Calculate distances:**
- \( a = \text{distance to vertex} = 1 \)
- \( c = \text{distance to focus} = 2 \)
- Use the relationship \( c^2 = a^2 + b^2 \) to find \( b \).
3. **Complete the equation:**
- Substitute \( h \), \( k \), \( a \), and \( b \) into the standard form of the hyperbola equation.
---
By understanding and solving these steps, you will be able to graph and write the equation of the given hyperbola.
Note: This educational exercise helps reinforce hyperbola properties and skills in deriving their equations.
---
#### Visual Aids:
If there are any accompanying graphs or diagrams, they should illustrate the center, foci, and vertices of the hyperbola with labeled coordinates to help visualize the problem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdcd4bca0-a10f-41b7-9fcd-0c13e56f6bb3%2F8cabaf61-f5a4-409a-8f07-09f9f590682c%2Fiyf56bm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
### Hyperbola Equation and Graphing
#### Problem Statement:
Find an equation for the hyperbola described. Graph the equation.
Given:
- Center at \( (8, -1) \)
- Focus at \( (10, -1) \)
- Vertex at \( (9, -1) \)
#### Task:
Write an equation for the hyperbola.
Equation Format (exact answers for each term, using fractions as needed):
\[ \frac{\Box}{\Box} - \frac{\Box}{\Box} = 1 \]
---
**Instructions:**
1. **Identify the Elements:**
- Center: \( (h, k) = (8, -1) \)
- Vertices: The distance from the center to a vertex (along the transverse axis) is the distance from \( (8, -1) \) to \( (9, -1) \), which is \( a \).
- Foci: The distance from the center to a focus (focal distance) is the distance from \( (8, -1) \) to \( (10, -1) \), which is \( c \).
2. **Calculate distances:**
- \( a = \text{distance to vertex} = 1 \)
- \( c = \text{distance to focus} = 2 \)
- Use the relationship \( c^2 = a^2 + b^2 \) to find \( b \).
3. **Complete the equation:**
- Substitute \( h \), \( k \), \( a \), and \( b \) into the standard form of the hyperbola equation.
---
By understanding and solving these steps, you will be able to graph and write the equation of the given hyperbola.
Note: This educational exercise helps reinforce hyperbola properties and skills in deriving their equations.
---
#### Visual Aids:
If there are any accompanying graphs or diagrams, they should illustrate the center, foci, and vertices of the hyperbola with labeled coordinates to help visualize the problem.
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