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The problem provides details for graphing and describing a hyperbola. Here is a step-by-step breakdown of how to approach it:

### Given Details:
- **Center (C)**: \( C(2, 2) \)
- **Focus (F)**: \( F(10, 2) \)
- **Vertex (V)**: \( V(5, 2) \)

### Step 1: Standard Form of Hyperbola Equation
Since the center and all given points have the same \(y\)-coordinate (i.e., 2), the transverse axis is horizontal. The equation of a horizontally oriented hyperbola centered at \(C(h, k)\) is:
\[
\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1
\]
Here, the center \(C(h, k)\) is \((2, 2)\), so the equation becomes:
\[
\frac{(x - 2)^2}{a^2} - \frac{(y - 2)^2}{b^2} = 1
\]

### Step 2: Determine \(a\)
The distance between the center and the vertex is \(a\). From the center \(C(2, 2)\) to the vertex \(V(5, 2)\), the distance is:
\[
a = |5 - 2| = 3
\]
Thus, \(a^2 = 9\).

### Step 3: Determine \(c\) (Distance from Center to Focus)
The distance between the center and the focus is \(c\). From the center \(C(2, 2)\) to the focus \(F(10, 2)\), the distance is:
\[
c = |10 - 2| = 8
\]
Thus, \(c^2 = 64\).

### Step 4: Find \(b^2\) Using the Relationship
For a hyperbola, the relationship between \(a\), \(b\), and \(c\) is given by:
\[
c^2 = a^2 + b^2
\]
Substituting the known values:
\[
64 = 9 + b^2
\]
Solving for \(b^2\):
\[
b^2 = 64 - 9 = 55
\]

### Step 5: Final Equation of the Hyperbola
Now that we know \(a^2 = 9\) and \(b^2 = 55\), the equation of the hyperbola is:
\[
\frac{(x - 2)^2}{9} - \frac{(y - 2)^2}{55} = 1
\]

### Step 6: Graph the Hyperbola
- **Center**: Plot the point \(C(2, 2)\).
- **Vertices**: Plot the vertices \(V(5, 2)\) and \(V(-1, 2)\) (since the distance is \(a = 3\) units to the left and right of the center).
- **Foci**: Plot the foci \(F(10, 2)\) and \(F(-6, 2)\) (since the distance is \(c = 8\) units to the left and right of the center).

The hyperbola will open left and right because the transverse axis is horizontal.

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### 5 Related Questions:
1. What is the difference between hyperbolas and ellipses in terms of their equations?
2. How does the orientation of a hyperbola change based on the location of its transverse axis?
3. What are asymptotes in a hyperbola, and how are they calculated?
4. How do you determine the equation of the conjugate axis of a hyperbola?
5. How do hyperbolas appear in real-world applications like satellite dishes or orbits?

### Tip:
When graphing conic sections like hyperbolas, always verify the orientation (horizontal vs. vertical) by checking the relative positions of the center, vertices, and foci.

Describe the given hyperbola completely using the given details: C(2, 2), F(10, 2), V(5, 2). Graph the curve.

Learn how to graph and describe the hyperbola with center (2, 2), focus (10, 2), and vertex (5, 2). This solution includes the equation of the hyperbola, distance calculations, and graphing steps. Suitable for high school students.

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