Math Problem Statement
Given equation x^2-4y^2-6x-8y-31=0. (1) write the equation in standard form. (2) find the vertices and foci. (3) find equations of asymptotes. (4) sketch the graph, using the center, vertices and asymptotes.
Solution
Here is the graph of the hyperbola, with the following elements highlighted:
- Center at , marked in red.
- Vertices at and , marked in green.
- Foci at and , marked in purple.
- Asymptotes represented by the dashed red lines, corresponding to the equations and .
Summary of Findings:
- Standard Form:
- Vertices: and
- Foci: and
- Asymptotes: and
Would you like more details or have any questions?
Here are 5 related questions to expand your understanding:
- How does the distance between the foci affect the shape of a hyperbola?
- How would the equation of the hyperbola change if the transverse axis were vertical instead of horizontal?
- How do the asymptotes help in sketching the graph of a hyperbola?
- What are the coordinates of the co-vertices for this hyperbola?
- How would you convert this hyperbola's equation into parametric form?
Tip: When identifying asymptotes of a hyperbola, the slopes are determined by the ratio , giving insight into the orientation and steepness of the asymptotes.
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Math Problem Analysis
Mathematical Concepts
Conic Sections
Hyperbola
Equation of a Hyperbola
Vertices
Foci
Asymptotes
Formulas
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Theorems
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Suitable Grade Level
Grades 11-12
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