Math Problem Statement
Solution
The challenge asks for the equation of a hyperbola that separates the blue points from the red points. Here's how we can solve this:
Step 1: Identify the orientation of the hyperbola
From the graph:
- The blue points are farther away horizontally from the red points.
- This suggests the hyperbola's branches open horizontally (left and right).
Thus, the equation of the hyperbola will take the standard form: where:
- is the center of the hyperbola,
- and are constants that determine the shape of the hyperbola.
Step 2: Locate the center of the hyperbola
The red points are close to the center of the hyperbola. From the graph:
- The red points seem symmetrically distributed around , meaning the center of the hyperbola is .
This simplifies the equation to:
Step 3: Determine and
-
Find :
- The vertices of the hyperbola lie along the -axis.
- From the graph, the vertices are at approximately and .
- The distance from the center to a vertex is , so .
-
Find :
- The red points at approximately and are near the "edges" of the box containing the hyperbola. These represent the co-vertices.
- The distance from the center to a co-vertex is , so .
Step 4: Write the equation
Substitute and into the equation:
This is the equation of the hyperbola.
Final Answer:
Would you like to go through additional details, clarifications, or examples?
Related Questions:
- What is the equation of a hyperbola that opens vertically instead of horizontally?
- How do we calculate the asymptotes of this hyperbola?
- How would the equation change if the center was at ?
- What is the relationship between , , and the focal distance ?
- How can we graph this hyperbola step-by-step?
Tip:
To verify your hyperbola equation, check the distances between the center, vertices, and co-vertices. Additionally, plot the asymptotes using the formula .
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Math Problem Analysis
Mathematical Concepts
Conic Sections
Hyperbolas
Graphing Equations
Formulas
Standard equation of a hyperbola opening horizontally: (x-h)^2/a^2 - (y-k)^2/b^2 = 1
Distance formula for vertices and co-vertices
Theorems
Properties of Hyperbolas
Symmetry of Conic Sections
Suitable Grade Level
Grades 10-12
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