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It seems like you're working with a problem involving a geometric shape, likely an ellipse, given the reference to a vertex and eccentricity. Let me help you break it down.

### Problem Details:
- **Vertices**: \( V(11, 4) \) and \( V'(-5, 4) \)
- **Eccentricity**: \( e = \frac{3}{5} \)

### Solution Steps:

1. **Identify the major axis**: 
   - The vertices have the same \( y \)-coordinate (4), so the major axis is horizontal.

2. **Find the center of the ellipse**:
   - The center is the midpoint between the two vertices:
   \[
   \text{Center} = \left(\frac{11 + (-5)}{2}, \frac{4 + 4}{2}\right) = \left(\frac{6}{2}, 4\right) = (3, 4)
   \]

3. **Find the distance between the center and each vertex**:
   - This distance, denoted as \( a \), is the semi-major axis length:
   \[
   a = \text{distance from } (3, 4) \text{ to } (11, 4) = |11 - 3| = 8
   \]

4. **Use the eccentricity to find \( c \)**:
   - The eccentricity is given by \( e = \frac{c}{a} \), where \( c \) is the distance from the center to each focus.
   \[
   c = e \times a = \frac{3}{5} \times 8 = \frac{24}{5} = 4.8
   \]

5. **Determine the foci**:
   - The foci will be at a distance \( c \) from the center along the major axis:
   \[
   \text{Foci: } \left(3 + 4.8, 4\right) = (7.8, 4) \text{ and } \left(3 - 4.8, 4\right) = (-1.8, 4)
   \]

6. **Calculate \( b \)**:
   - The relationship \( b^2 = a^2 - c^2 \) allows us to find the semi-minor axis:
   \[
   b^2 = 8^2 - 4.8^2 = 64 - 23.04 = 40.96
   \]
   \[
   b = \sqrt{40.96} \approx 6.4
   \]

7. **Write the equation of the ellipse**:
   - The equation of an ellipse centered at \( (h, k) \) with horizontal major axis is:
   \[
   \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1
   \]
   Substituting the known values:
   \[
   \frac{(x - 3)^2}{64} + \frac{(y - 4)^2}{40.96} = 1
   \]

### Conclusion:
The ellipse has its center at \( (3, 4) \), a semi-major axis length of 8, a semi-minor axis length of approximately 6.4, and foci located at \( (7.8, 4) \) and \( (-1.8, 4) \). The equation of the ellipse is:

\[
\frac{(x - 3)^2}{64} + \frac{(y - 4)^2}{40.96} = 1
\]

Would you like further details on any specific part of the solution, or do you have any questions?

### Related Questions:
1. What is the geometric significance of the eccentricity of an ellipse?
2. How does the distance between the foci affect the shape of the ellipse?
3. Can the method used here be applied to an ellipse with a vertical major axis?
4. How would the equation of the ellipse change if the center were moved to the origin?
5. What happens to the ellipse if the eccentricity is increased to 1?

### Tip:
When solving problems involving ellipses, always identify the major and minor axes first, as this will guide you in correctly setting up the equation.

Learn how to find the vertices and eccentricity of an ellipse with a detailed step-by-step solution. This problem involves identifying vertices, calculating distances, determining eccentricity, and writing the equation of the ellipse. Suitable for advanced high school students.

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