Ellipse with center (3,-1)

Focus (3,3) (3,-5)

Vertex (3,4) (3,-6)

To analyze the given ellipse:

### 1. **Identifying the Elements of the Ellipse**:
   - **Center**: \((3, -1)\)
   - **Foci**: \((3, 3)\) and \((3, -5)\)
   - **Vertices**: \((3, 4)\) and \((3, -6)\)

### 2. **Understanding the Structure**:
   - Since both the foci and vertices have the same x-coordinate (3), the ellipse is **vertically oriented**.
   - The **center** of the ellipse is at \((3, -1)\), so this point is equidistant from the vertices and the foci.

### 3. **Key Parameters**:
   - **Distance between the center and a vertex** (which is the semi-major axis \(a\)):
     \[
     a = \text{Distance from } (3, -1) \text{ to } (3, 4) = |4 - (-1)| = 5
     \]
   - **Distance between the center and a focus** (which is \(c\)):
     \[
     c = \text{Distance from } (3, -1) \text{ to } (3, 3) = |3 - (-1)| = 4
     \]
   - **The semi-minor axis \(b\)** can be found using the relationship \(b^2 = a^2 - c^2\):
     \[
     b^2 = 5^2 - 4^2 = 25 - 16 = 9 \quad \text{so} \quad b = 3
     \]

### 4. **Equation of the Ellipse**:
   Since the ellipse is vertically oriented, the equation is:
   \[
   \frac{(x - 3)^2}{b^2} + \frac{(y + 1)^2}{a^2} = 1
   \]
   Substituting \(a = 5\) and \(b = 3\):
   \[
   \frac{(x - 3)^2}{9} + \frac{(y + 1)^2}{25} = 1
   \]

This is the equation of the ellipse.

Would you like more details or have any questions?

### Related Questions:
1. How do you find the equation of an ellipse given different sets of points?
2. What is the relationship between the foci and the shape of an ellipse?
3. How do you calculate the eccentricity of an ellipse?
4. What are some practical applications of ellipses in real life?
5. How do the lengths of the semi-major and semi-minor axes affect the graph of an ellipse?

### Tip:
Always ensure the orientation of the ellipse (vertical or horizontal) before writing its equation.

Explore the properties and equation of an ellipse with a given center at (3, -1), foci at (3, 3) and (3, -5), and vertices at (3, 4) and (3, -6). Learn how to derive the equation and understand the geometric properties of the ellipse. Suitable for high school students and enthusiasts of geometry and conic sections.

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