An equation of a hyperbola is given. 7²-2=1 (a) Find the vertices, foci, and asymptotes of the hyperbola. (Enter your asymptotes as a comma-separated list of equations.) vertex (x, y) = ( (smaller y-value) vertex (x, y) = (larger y-value) focus (x, y) = (smaller y-value) focus (x, y) = (larger y-value) asymptotes (b) Determine the length of the transverse axis.
An equation of a hyperbola is given. 7²-2=1 (a) Find the vertices, foci, and asymptotes of the hyperbola. (Enter your asymptotes as a comma-separated list of equations.) vertex (x, y) = ( (smaller y-value) vertex (x, y) = (larger y-value) focus (x, y) = (smaller y-value) focus (x, y) = (larger y-value) asymptotes (b) Determine the length of the transverse axis.
Calculus: Early Transcendentals
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Chapter1: Functions And Models
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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 Analysis and Calculations**
**Problem Statement:**
An equation of a hyperbola is given:
\[ \frac{y^2}{25} - \frac{x^2}{9} = 1 \]
**Tasks:**
**(a) Find the vertices, foci, and asymptotes of the hyperbola. (Enter your asymptotes as a comma-separated list of equations.)**
- **Vertex (x, y) = (_________)** (smaller y-value)
- **Vertex (x, y) = (_________)** (larger y-value)
- **Focus (x, y) = (_________)** (smaller y-value)
- **Focus (x, y) = (_________)** (larger y-value)
- **Asymptotes**: _________
**(b) Determine the length of the transverse axis.**
- The length of the transverse axis: _________
**Explanation of Terms:**
1. **Vertices**: The points where the hyperbola intersects its transverse axis.
2. **Foci**: The points from which the sum of the distances to any point on the hyperbola is constant.
3. **Asymptotes**: The lines that the hyperbola approaches but never meets.
4. **Transverse Axis**: The axis that passes through the vertices of the hyperbola. The length of the transverse axis is the distance between the vertices.
This interactive problem requires students to compute specific features of a hyperbola from its standard form equation. For a hyperbola given by \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\):
- The vertices are located at \((0, \pm a)\).
- The foci can be found using \((0, \pm c)\) where \(c = \sqrt{a^2 + b^2}\).
- The equations of the asymptotes for this hyperbola are \(y = \pm \frac{a}{b}x\).
- The length of the transverse axis is \(2a\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3f29774d-39d7-4c90-90ba-0f0924899c12%2F56c5a290-ab99-4c0d-a505-f6b8b4a16b79%2Fj7cfj49_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Hyperbola Analysis and Calculations**
**Problem Statement:**
An equation of a hyperbola is given:
\[ \frac{y^2}{25} - \frac{x^2}{9} = 1 \]
**Tasks:**
**(a) Find the vertices, foci, and asymptotes of the hyperbola. (Enter your asymptotes as a comma-separated list of equations.)**
- **Vertex (x, y) = (_________)** (smaller y-value)
- **Vertex (x, y) = (_________)** (larger y-value)
- **Focus (x, y) = (_________)** (smaller y-value)
- **Focus (x, y) = (_________)** (larger y-value)
- **Asymptotes**: _________
**(b) Determine the length of the transverse axis.**
- The length of the transverse axis: _________
**Explanation of Terms:**
1. **Vertices**: The points where the hyperbola intersects its transverse axis.
2. **Foci**: The points from which the sum of the distances to any point on the hyperbola is constant.
3. **Asymptotes**: The lines that the hyperbola approaches but never meets.
4. **Transverse Axis**: The axis that passes through the vertices of the hyperbola. The length of the transverse axis is the distance between the vertices.
This interactive problem requires students to compute specific features of a hyperbola from its standard form equation. For a hyperbola given by \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\):
- The vertices are located at \((0, \pm a)\).
- The foci can be found using \((0, \pm c)\) where \(c = \sqrt{a^2 + b^2}\).
- The equations of the asymptotes for this hyperbola are \(y = \pm \frac{a}{b}x\).
- The length of the transverse axis is \(2a\).
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