conic in standard form.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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![**Exploring Conic Sections: Identifying and Sketching**
In this exercise, we'll find the equation of a conic section graph in its standard form. The task involves identifying aspects such as the type of conic, its center, vertex, co-vertex, focus (or foci), major axis, minor axis, and the values of \(a^2\), \(b^2\), and \(c^2\). For hyperbolas, you'll also find the asymptotes and sketch the graph.
**Problem:**
Given the equation:
\[ 4x^2 + 2y^2 + 24x + 12y + 46 = 0 \]
**Instructions:**
1. **Identify the Conic Type:** Analyze the equation to determine whether it represents a circle, ellipse, hyperbola, or parabola.
2. **Find the Components:**
- **Center:** Calculate the center coordinates.
- **Vertex and Co-Vertex:** Determine their positions.
- **Focus (Foci):** Identify and locate the foci.
- **Axes:** Distinguish between the major and minor axes.
- **Coefficients:** Compute \(a^2\), \(b^2\), and \(c^2\).
3. **For Hyperbolas:** Determine the asymptotes.
4. **Sketch the Graph:** Visualize and draw the graph based on the identified components.
This process will help in understanding the geometry and algebra involved in conic sections.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd21d8c83-5470-430d-a8ec-b76a155fcd97%2F10f87261-f679-4e37-a76e-6c9d84e08c60%2Fp04viao_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Exploring Conic Sections: Identifying and Sketching**
In this exercise, we'll find the equation of a conic section graph in its standard form. The task involves identifying aspects such as the type of conic, its center, vertex, co-vertex, focus (or foci), major axis, minor axis, and the values of \(a^2\), \(b^2\), and \(c^2\). For hyperbolas, you'll also find the asymptotes and sketch the graph.
**Problem:**
Given the equation:
\[ 4x^2 + 2y^2 + 24x + 12y + 46 = 0 \]
**Instructions:**
1. **Identify the Conic Type:** Analyze the equation to determine whether it represents a circle, ellipse, hyperbola, or parabola.
2. **Find the Components:**
- **Center:** Calculate the center coordinates.
- **Vertex and Co-Vertex:** Determine their positions.
- **Focus (Foci):** Identify and locate the foci.
- **Axes:** Distinguish between the major and minor axes.
- **Coefficients:** Compute \(a^2\), \(b^2\), and \(c^2\).
3. **For Hyperbolas:** Determine the asymptotes.
4. **Sketch the Graph:** Visualize and draw the graph based on the identified components.
This process will help in understanding the geometry and algebra involved in conic sections.
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