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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please can I have step by step and formula with an explanation
![**Finding the Equation of a Hyperbola**
To find the equation of a hyperbola with the specified conditions:
1. **Vertices**: (0, ±2)
2. **Foci**: (0, ±5)
### General Form of a Vertical Hyperbola
The general equation for a vertical hyperbola centered at the origin \((0, 0)\) is:
\[
\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1
\]
### Determining Parameters
- **Distance to Vertices (\(a\))**:
- Since the vertices are \( (0, ±2) \), \( a = 2 \).
- **Distance to Foci (\(c\))**:
- Since the foci are \( (0, ±5) \), \( c = 5 \).
### Relationship between \(a\), \(b\), and \(c\)
For hyperbolas, the relationship between \(a\), \(b\), and \(c\) is given by:
\[
c^2 = a^2 + b^2
\]
Plugging in the values:
\[
5^2 = 2^2 + b^2
\]
\[
25 = 4 + b^2
\]
\[
b^2 = 21
\]
### Equation of the Hyperbola
Using the values \(a^2 = 4\) and \(b^2 = 21\), the equation becomes:
\[
\frac{y^2}{4} - \frac{x^2}{21} = 1
\]
This is the required equation of the hyperbola.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4d62cc64-0378-4638-9893-20a25cf6ef2f%2Fe6a2a002-01c5-4c4c-84c3-e815d07971dd%2Fyhg3cxb_processed.png&w=3840&q=75)
Transcribed Image Text:**Finding the Equation of a Hyperbola**
To find the equation of a hyperbola with the specified conditions:
1. **Vertices**: (0, ±2)
2. **Foci**: (0, ±5)
### General Form of a Vertical Hyperbola
The general equation for a vertical hyperbola centered at the origin \((0, 0)\) is:
\[
\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1
\]
### Determining Parameters
- **Distance to Vertices (\(a\))**:
- Since the vertices are \( (0, ±2) \), \( a = 2 \).
- **Distance to Foci (\(c\))**:
- Since the foci are \( (0, ±5) \), \( c = 5 \).
### Relationship between \(a\), \(b\), and \(c\)
For hyperbolas, the relationship between \(a\), \(b\), and \(c\) is given by:
\[
c^2 = a^2 + b^2
\]
Plugging in the values:
\[
5^2 = 2^2 + b^2
\]
\[
25 = 4 + b^2
\]
\[
b^2 = 21
\]
### Equation of the Hyperbola
Using the values \(a^2 = 4\) and \(b^2 = 21\), the equation becomes:
\[
\frac{y^2}{4} - \frac{x^2}{21} = 1
\]
This is the required equation of the hyperbola.
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