equation of the graph for each conic in standard form. e co-vertex the focus (foci)
equation of the graph for each conic in standard form. e co-vertex the focus (foci)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Title: Understanding Conic Sections: Analyzing and Graphing**
**Objective:**
Learn how to find the equation of conic sections in standard form and identify key features such as the center, vertex, co-vertex, focus (foci), major axis, minor axis, and asymptotes. Practice sketching the graph for a given equation.
---
**Problem Statement:**
Find the equation of the graph for each conic in standard form. Identify the conic, the center, the vertex, the co-vertex, the focus (foci), major axis, minor axis, \(a^2\), \(b^2\), and \(c^2\). For a hyperbola, find the asymptotes. Sketch the graph.
Given equation:
\[ 2x^2 + 3y^2 + 6y - 4x - 1 = 0 \]
**Solution Steps:**
1. **Identify the Type of Conic:**
Start by rearranging the equation to detect whether it represents an ellipse, parabola, or hyperbola.
2. **Convert to Standard Form:**
Rearrange terms to bring the equation into standard form. Complete the square if necessary for both \(x\) and \(y\) terms.
3. **Calculate the Parameters:**
- Determine the center of the conic section.
- Identify the vertices and co-vertices.
- Calculate the foci using the relationships between \(a^2\), \(b^2\), and \(c^2\).
- For hyperbolas, determine the equations of the asymptotes.
4. **Sketch the Graph:**
Use the identified parameters to accurately sketch the conic section.
**Note:** There are no graphs or diagrams included with this explanation, as the focus is on the theoretical framework and calculations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd21d8c83-5470-430d-a8ec-b76a155fcd97%2F2daaf396-203c-4df4-b343-f705f92d5d83%2Fci9hgdf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Understanding Conic Sections: Analyzing and Graphing**
**Objective:**
Learn how to find the equation of conic sections in standard form and identify key features such as the center, vertex, co-vertex, focus (foci), major axis, minor axis, and asymptotes. Practice sketching the graph for a given equation.
---
**Problem Statement:**
Find the equation of the graph for each conic in standard form. Identify the conic, the center, the vertex, the co-vertex, the focus (foci), major axis, minor axis, \(a^2\), \(b^2\), and \(c^2\). For a hyperbola, find the asymptotes. Sketch the graph.
Given equation:
\[ 2x^2 + 3y^2 + 6y - 4x - 1 = 0 \]
**Solution Steps:**
1. **Identify the Type of Conic:**
Start by rearranging the equation to detect whether it represents an ellipse, parabola, or hyperbola.
2. **Convert to Standard Form:**
Rearrange terms to bring the equation into standard form. Complete the square if necessary for both \(x\) and \(y\) terms.
3. **Calculate the Parameters:**
- Determine the center of the conic section.
- Identify the vertices and co-vertices.
- Calculate the foci using the relationships between \(a^2\), \(b^2\), and \(c^2\).
- For hyperbolas, determine the equations of the asymptotes.
4. **Sketch the Graph:**
Use the identified parameters to accurately sketch the conic section.
**Note:** There are no graphs or diagrams included with this explanation, as the focus is on the theoretical framework and calculations.
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