Math Problem Statement
Solution
To solve this, let’s analyze the given equation:
Steps to Identify the Conic Section:
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Compare the coefficients of and :
- The coefficients are and , both positive, which suggests it might be an ellipse.
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Simplify the equation into standard form: Let's complete the square for and .
Completing the square for :
Complete the square: Substituting back:
Completing the square for :
Complete the square: Substituting back:
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Rewriting the equation: Substitute completed squares into the equation: Simplify:
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Divide through by 64 to standardize:
This is the equation of an ellipse.
Final Answer:
The given equation represents an ellipse.
Do you need further clarification or details? Here are 5 related questions for further exploration:
- What are the major and minor axes of the ellipse derived above?
- How do you determine the center of a conic section from its equation?
- What distinguishes an ellipse from a hyperbola in their general equations?
- Can you explain the process of completing the square in more detail?
- How would this equation look if it were a circle instead of an ellipse?
Tip: Always check the signs and relative magnitudes of the squared terms’ coefficients to quickly narrow down the type of conic section.
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Math Problem Analysis
Mathematical Concepts
Conic Sections
Ellipse
Completing the Square
Formulas
Equation of ellipse: (x-h)^2/a^2 + (y-k)^2/b^2 = 1
Completing the square
Theorems
Properties of Conic Sections
Suitable Grade Level
College-level Algebra