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what are the asymptotes, standard form, general equation and E1,E2,E3,E5

To describe the given hyperbola completely and determine its elements such as asymptotes, standard and general form equations, and the eccentricity values (E1, E2, E3, E5), let's analyze the information provided:

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

### Analysis:
1. **Orientation**: Since the focus and vertex have the same \(y\)-coordinates but different \(x\)-coordinates, the hyperbola is horizontal.
2. **Distance Calculations**:
   - Distance from center to vertex (a): \(|5 - 2| = 3\)
   - Distance from center to focus (c): \(|10 - 2| = 8\)

### Equations:
- **Standard Form**: For a horizontal hyperbola centered at \((h, k)\) with distances \(a\) and \(c\) (where \(c > a\)), the standard form equation is:
  \[
  \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1
  \]
  Here, \(b^2 = c^2 - a^2\).
- **Calculating \(b^2\)**:
  \[
  b^2 = 8^2 - 3^2 = 64 - 9 = 55
  \]
- **Standard Form Equation**:
  \[
  \frac{(x - 2)^2}{9} - \frac{(y - 2)^2}{55} = 1
  \]

- **Asymptotes**: The equations of the asymptotes for a horizontal hyperbola are given by:
  \[
  y - k = \pm \frac{b}{a}(x - h)
  \]
  Plugging in our values:
  \[
  y - 2 = \pm \frac{\sqrt{55}}{3} (x - 2)
  \]

- **General Equation**: Expanding the standard form:
  \[
  \frac{(x - 2)^2}{9} - \frac{(y - 2)^2}{55} = 1
  \]
  Multiply through by \(9 \times 55\):
  \[
  55(x - 2)^2 - 9(y - 2)^2 = 495
  \]
  Which simplifies to:
  \[
  55x^2 - 220x + 55y^2 - 18y^2 + 396 = 495
  \]

### Eccentricities (E1, E2, E3, E5):
- **Eccentricity (e)**: \( e = \frac{c}{a} = \frac{8}{3} \)
- The other values (E1, E2, E3, E5) are not standard terms used in describing conics, and clarification would be needed on what these refer to.

Would you like further details on any of these calculations, or do you have any specific questions about the hyperbola? Here are some additional topics you might consider:
1. Deriving the formula for the directrix of the hyperbola.
2. Explaining the significance of the eccentricity value.
3. Discussing how to plot the hyperbola on the graph paper shown in the image.
4. Calculating the lengths of the semi-major and semi-minor axes.
5. Describing how the properties of the hyperbola change with different values of \(a\) and \(b\).

**Tip**: Always verify the location of the foci and vertices by calculating distances from the center to ensure consistency in the elements of the hyperbola's equation.

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

Learn how to describe a hyperbola with center C(2,2), focus F(10,2), and vertex V(5,2). This involves calculating the standard form equation, asymptotes, and other elements. Perfect for students in Grades 10-12 studying conic sections.

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