Math Problem Statement

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

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: (xh)2a2(yk)2b2=1\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 where:

  • (h,k)(h, k) is the center of the hyperbola,
  • a2a^2 and b2b^2 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 (0,0)(0, 0), meaning the center of the hyperbola is (h,k)=(0,0)(h, k) = (0, 0).

This simplifies the equation to: x2a2y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1


Step 3: Determine a2a^2 and b2b^2

  1. Find a2a^2:

    • The vertices of the hyperbola lie along the xx-axis.
    • From the graph, the vertices are at approximately (5,0)(-5, 0) and (5,0)(5, 0).
    • The distance from the center to a vertex is a=5a = 5, so a2=25a^2 = 25.
  2. Find b2b^2:

    • The red points at approximately (0,2)(0, 2) and (0,2)(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=2b = 2, so b2=4b^2 = 4.

Step 4: Write the equation

Substitute a2=25a^2 = 25 and b2=4b^2 = 4 into the equation: x225y24=1\frac{x^2}{25} - \frac{y^2}{4} = 1

This is the equation of the hyperbola.


Final Answer:

x225y24=1\boxed{\frac{x^2}{25} - \frac{y^2}{4} = 1}

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


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)(0,0)(h, k) \neq (0, 0)?
  4. What is the relationship between aa, bb, and the focal distance cc?
  5. 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 y=±baxy = \pm \frac{b}{a}x.

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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