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We are tasked with determining the center of an ellipse given certain points related to it. Let's break down the information provided:

1. **Focus** at point \((-3, 5)\).
2. **Co-vertex** at point \((2, -2)\).
3. The length of the semi-minor axis is longer than the focal distance.

### Step-by-step approach:

- **General form of an ellipse equation**:
  The general equation of an ellipse is:
  \[
  \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1
  \]
  where \((h, k)\) is the center of the ellipse, \(a\) is the semi-major axis length, and \(b\) is the semi-minor axis length.

- **Focus of the ellipse**: 
  The foci of an ellipse are located along the major axis, at a distance \(c\) from the center, where \(c = \sqrt{a^2 - b^2}\).

- **Co-vertex of the ellipse**:
  The co-vertices are located along the minor axis, and their distance from the center is \(b\) (the semi-minor axis length).

- **Conditions**:
  - The length of the semi-minor axis is greater than the focal distance, meaning \(b > c\).
  - The center is equidistant from both the focus and the co-vertex.

### Finding the center:
The center of the ellipse must lie between the focus and the co-vertex. Given that the focal point is at \((-3, 5)\) and the co-vertex is at \((2, -2)\), the center is likely to be the midpoint of these two points.

Using the midpoint formula:
\[
\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]

Substitute \((-3, 5)\) and \((2, -2)\):
\[
\left( \frac{-3 + 2}{2}, \frac{5 + (-2)}{2} \right) = \left( \frac{-1}{2}, \frac{3}{2} \right) = \left( -\frac{1}{2}, \frac{3}{2} \right)
\]

Thus, the center of the ellipse is at \(\left( -\frac{1}{2}, \frac{3}{2} \right)\).

Would you like more details on this process or further explanation?

Here are some additional questions to deepen your understanding:

1. What is the difference between the semi-major and semi-minor axes in an ellipse?
2. How do you calculate the focal distance of an ellipse?
3. Why is the length of the semi-minor axis greater than the focal distance relevant to the shape of the ellipse?
4. How can the relationship between the focus and co-vertex help determine the center of the ellipse?
5. What is the significance of the midpoint in determining the center of conic sections?

**Tip**: Always check the symmetry of an ellipse; the midpoint of key points such as the foci and vertices is often the center.

What is the center of the ellipse with a focus at (-3, 5) and a co-vertex at (2, -2) if the length of its semi-minor axis is longer than its focal distance?

Determine the center of an ellipse with a focus at (-3, 5) and a co-vertex at (2, -2). Explore how to use the midpoint formula and key properties of ellipses in this detailed geometric problem. Suitable for Grades 11-12 or early college-level students.

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