Math Problem Statement
What is the standard form equation of the hyperbola that has vertices (0,±2) and foci (0,±27–√)
?
Enter the exact equation.
Include a multiplication sign between symbols. For example, x*5–√ .
Solution
To determine the equation of the hyperbola, let's analyze the given information step by step.
Step 1: Identify the type of hyperbola
The hyperbola has vertices at , meaning it is centered at and opens along the -axis. This makes it a vertical hyperbola.
The standard form of a vertical hyperbola is:
Step 2: Find
The distance from the center to the vertices is . Here, , so:
Step 3: Use the distance between the foci to find
The distance from the center to the foci is . Here, , so:
Step 4: Find using the relationship
The relationship between , , and in a hyperbola is: Substitute the known values: Solve for :
Step 5: Write the equation
Substitute and into the standard form equation:
Final Answer:
The standard form of the hyperbola is:
Would you like further details or clarifications?
Related Questions:
- How do you derive the standard form of a hyperbola?
- What are the equations of asymptotes for this hyperbola?
- How do you graph a hyperbola given its equation?
- Can you find the eccentricity of this hyperbola?
- What is the significance of , , and in the hyperbola's equation?
Tip:
Always verify whether the hyperbola opens horizontally or vertically before using its standard
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Math Problem Analysis
Mathematical Concepts
Analytic Geometry
Conic Sections
Hyperbolas
Formulas
Standard form of a vertical hyperbola: (y^2 / a^2) - (x^2 / b^2) = 1
Relationship: c^2 = a^2 + b^2
Theorems
Definition of Hyperbolas
Distance Formula for Conic Sections
Suitable Grade Level
Grades 10-12
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