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We are tasked with finding the standard equation of an ellipse based on the given conditions:

- **Focus:** \((-4, 8)\)
- **Co-vertex:** \((-1, 5)\)
- **Horizontal major axis**

### Step-by-step approach:

1. **Determine the center** of the ellipse:
   - The center of an ellipse lies at the midpoint between the two co-vertices (since the major axis is horizontal).
   - The co-vertex is given as \((-1, 5)\), and since the major axis is horizontal, the center must lie on the same horizontal line, likely at \((-1, 8)\) because this is consistent with the vertical positioning of the focus and co-vertex.

2. **Identify \(c\) (distance from the center to the focus):**
   - The focus is at \((-4, 8)\), and the center is at \((-1, 8)\). Since the distance between these two points is along the horizontal axis, \(c = |-4 - (-1)| = 3\).

3. **Identify \(b\) (distance from the center to the co-vertex):**
   - The co-vertex is at \((-1, 5)\), and the center is at \((-1, 8)\). This distance is along the vertical axis, so \(b = |5 - 8| = 3\).

4. **Use the relationship \(a^2 = b^2 + c^2\) to find \(a\) (semi-major axis):**
   - Since \(c = 3\) and \(b = 3\), we can substitute these values into the equation:
     \[
     a^2 = b^2 + c^2 = 3^2 + 3^2 = 9 + 9 = 18
     \]
     Thus, \(a = \sqrt{18} = 3\sqrt{2}\).

5. **Write the standard equation** of the ellipse:
   - The standard form of an ellipse with a horizontal major axis is:
     \[
     \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1
     \]
     where \((h, k)\) is the center, \(a\) is the semi-major axis, and \(b\) is the semi-minor axis.
   - Substituting the known values \((h, k) = (-1, 8)\), \(a^2 = 18\), and \(b^2 = 9\), we get the equation:
     \[
     \frac{(x + 1)^2}{18} + \frac{(y - 8)^2}{9} = 1
     \]

### Final equation of the ellipse:
\[
\frac{(x + 1)^2}{18} + \frac{(y - 8)^2}{9} = 1
\]

Let me know if you'd like more details or have any questions! Here are some related questions to explore further:

1. How do we determine whether an ellipse has a horizontal or vertical major axis?
2. What is the relationship between the foci and the center of an ellipse?
3. Can you derive the equation of an ellipse if given only the vertices and the length of the minor axis?
4. How does the standard equation of a circle compare to that of an ellipse?
5. What is the significance of the foci in the geometry of an ellipse?

**Tip:** The values \(a\), \(b\), and \(c\) in an ellipse are always related by the equation \(a^2 = b^2 + c^2\), where \(a\) is the length of the semi-major axis, \(b\) the semi-minor axis, and \(c\) the distance from the center to each focus.

Find the standard equation of the ellipse with the given condition: Focus (-4, 8), Co-vertex (-1, 5), Horizontal Major Axis.

Find the standard equation of an ellipse with a focus at (-4, 8) and a co-vertex at (-1, 5). This problem involves calculating the center, distances to the foci, and using the relationship a^2 = b^2 + c^2 to solve for the equation. Suitable for students in Grades 10-12.

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