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You are asked to find the equation of an ellipse with the following dataLet's analyze the problem from the image:

You are asked to find the equation of an ellipse with the following 
- The center of the ellipse is \((5, -4)\).
- The length of the major axis is 10.
- One endpoint of the minor axis is at \((6, -4)\).

### Step-by-Step Solution:

1. **Center of the ellipse**: The center is at \((h, k) = (5, -4)\).

2. **Length of the major axis**: The major axis has a total length of 10, which means that half of this length (semi-major axis) is 5. Therefore, \(a = 5\).

3. **Orientation of the axes**: Since one endpoint of the minor axis is given as \((6, -4)\), and the center is \((5, -4)\), the minor axis is horizontal. The distance between the center and this endpoint is 1, so the semi-minor axis \(b = 1\).

4. **Equation of the ellipse**: The general form of the equation for an ellipse with center at \((h, k)\), semi-major axis \(a\), and semi-minor axis \(b\) is:

   \[
   \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1
   \]

   Substituting the values \(h = 5\), \(k = -4\), \(a = 5\), and \(b = 1\):

   \[
   \frac{(x - 5)^2}{5^2} + \frac{(y + 4)^2}{1^2} = 1
   \]

   Simplifying:

   \[
   \frac{(x - 5)^2}{25} + (y + 4)^2 = 1
   \]

This is the equation of the ellipse.

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Would you like further details or explanations on any step?

Here are some related questions to explore:
1. What is the geometric significance of the foci in an ellipse?
2. How do we determine the eccentricity of an ellipse?
3. How would the equation change if the major axis were along the y-axis instead of the x-axis?
4. What happens to the equation if the ellipse is a circle (i.e., \(a = b\))?
5. Can you find the foci for this specific ellipse?

**Tip:** For ellipses, always double-check which axis is the major and which is the minor to avoid confusion with the lengths.

Hallar una ecuación de la elipse con centro en (5, -4), eje mayor de longitud 10 y un extremo del eje menor en (6, -4).

Find the equation of an ellipse with a center at (5, -4), a major axis length of 10, and a minor axis endpoint at (6, -4). Step-by-step guide on solving ellipses using geometric principles.

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