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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:

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### 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:
\[
\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1
\]
where:
- \((h, k)\) is the center of the hyperbola,
- \(a^2\) and \(b^2\) are constants that determine the shape of the hyperbola.

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### 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 \((0, 0)\), meaning the center of the hyperbola is \((h, k) = (0, 0)\).

This simplifies the equation to:
\[
\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1
\]

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### Step 3: Determine \(a^2\) and \(b^2\)
1. **Find \(a^2\):**
   - The vertices of the hyperbola lie along the \(x\)-axis.
   - From the graph, the vertices are at approximately \((-5, 0)\) and \((5, 0)\).
   - The distance from the center to a vertex is \(a = 5\), so \(a^2 = 25\).

2. **Find \(b^2\):**
   - The red points at approximately \((0, 2)\) and \((0, -2)\) 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 \(b = 2\), so \(b^2 = 4\).

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### Step 4: Write the equation
Substitute \(a^2 = 25\) and \(b^2 = 4\) into the equation:
\[
\frac{x^2}{25} - \frac{y^2}{4} = 1
\]

This is the equation of the hyperbola.

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### Final Answer:
\[
\boxed{\frac{x^2}{25} - \frac{y^2}{4} = 1}
\]

Would you like to go through additional details, clarifications, or examples?

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### Related Questions:
1. What is the equation of a hyperbola that opens vertically instead of horizontally?
2. How do we calculate the asymptotes of this hyperbola?
3. How would the equation change if the center was at \((h, k) \neq (0, 0)\)?
4. What is the relationship between \(a\), \(b\), and the focal distance \(c\)?
5. How can we graph this hyperbola step-by-step?

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### Tip:
To verify your hyperbola equation, check the distances between the center, vertices, and co-vertices. Additionally, plot the asymptotes using the formula \(y = \pm \frac{b}{a}x\).

Write the equation of a hyperbola whose graph separates the blue points from the red points.

Learn how to determine the equation of a hyperbola that separates two sets of points on a graph. This problem involves identifying the center, vertices, and co-vertices of the hyperbola and applying its standard equation. Ideal for high school students in Grades 10-12 studying conic sections.

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