12. Look for Relationships Explain how to find the value of b in the equation of a hyperbola, given one focus at (0, 1) and one vertex at (0, 0.5).
12. Look for Relationships Explain how to find the value of b in the equation of a hyperbola, given one focus at (0, 1) and one vertex at (0, 0.5).
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
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12 please
![### Hyperbolas
#### Definition:
A hyperbola is the set of all points \(P\) such that the difference of the distances from any point \(P\) to two fixed points is constant.
#### Graphs:
**Horizontal Hyperbola:**
- The graph shows a horizontal hyperbola with its transverse axis along the x-axis.
- It has two vertices located at \((\pm a, 0)\).
- Asymptotes are the lines that the hyperbola approaches but never meets. They are shown in blue and have the equations \(y = \pm \frac{b}{a}x\).
- The foci, represented by \(F_1\) and \(F_2\), are located along the x-axis at \((\pm c, 0)\) where \(c = \sqrt{a^2 + b^2}\).
**Vertical Hyperbola:**
- This graph represents a vertical hyperbola with its transverse axis along the y-axis.
- It has two vertices located at \((0, \pm a)\).
- The asymptotes, shown in blue, have the equations \(y = \pm \frac{a}{b}x\).
- The foci, represented by \(F_1\) and \(F_2\), are located along the y-axis at \((0, \pm c)\) where \(c = \sqrt{a^2 + b^2}\).
#### Equation:
**Horizontal:**
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
- Vertices: \((\pm a, 0)\)
- Asymptotes: \(y = \pm \frac{b}{a}x\)
- Foci: \((\pm c, 0)\), where \(c = \sqrt{a^2 + b^2}\)
**Vertical:**
\[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \]
- Vertices: \((0, \pm a)\)
- Asymptotes: \(y = \pm \frac{a}{b}x\)
- Foci: \((0, \pm c)\), where \(c = \sqrt{a^2 + b^2}\)
#### Do You Understand?
1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6e05d8d1-3997-4023-8636-4176c027f4f7%2Ff2643d7b-1b81-4e8e-ac72-122c8eb71768%2Fb44rzys_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Hyperbolas
#### Definition:
A hyperbola is the set of all points \(P\) such that the difference of the distances from any point \(P\) to two fixed points is constant.
#### Graphs:
**Horizontal Hyperbola:**
- The graph shows a horizontal hyperbola with its transverse axis along the x-axis.
- It has two vertices located at \((\pm a, 0)\).
- Asymptotes are the lines that the hyperbola approaches but never meets. They are shown in blue and have the equations \(y = \pm \frac{b}{a}x\).
- The foci, represented by \(F_1\) and \(F_2\), are located along the x-axis at \((\pm c, 0)\) where \(c = \sqrt{a^2 + b^2}\).
**Vertical Hyperbola:**
- This graph represents a vertical hyperbola with its transverse axis along the y-axis.
- It has two vertices located at \((0, \pm a)\).
- The asymptotes, shown in blue, have the equations \(y = \pm \frac{a}{b}x\).
- The foci, represented by \(F_1\) and \(F_2\), are located along the y-axis at \((0, \pm c)\) where \(c = \sqrt{a^2 + b^2}\).
#### Equation:
**Horizontal:**
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
- Vertices: \((\pm a, 0)\)
- Asymptotes: \(y = \pm \frac{b}{a}x\)
- Foci: \((\pm c, 0)\), where \(c = \sqrt{a^2 + b^2}\)
**Vertical:**
\[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \]
- Vertices: \((0, \pm a)\)
- Asymptotes: \(y = \pm \frac{a}{b}x\)
- Foci: \((0, \pm c)\), where \(c = \sqrt{a^2 + b^2}\)
#### Do You Understand?
1
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