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To find the standard form of the equation of an ellipse, let’s go through the information provided:

### Given Information:
- **Major axis**: Vertical with length 22.
- **Minor axis**: Length 10.
- **Center**: \((-5, -2)\).

### Solution:
1. **Center of the Ellipse**:
   The center of the ellipse is at \((h, k) = (-5, -2)\).

2. **Lengths of the Axes**:
   - The **major axis** length is 22, so the **semi-major axis** \(a = \frac{22}{2} = 11\).
   - The **minor axis** length is 10, so the **semi-minor axis** \(b = \frac{10}{2} = 5\).

3. **Equation Form**:
   Since the major axis is vertical, the standard form of the ellipse’s equation is:
   \[
   \frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1
   \]
   Substituting \(h = -5\), \(k = -2\), \(a = 11\), and \(b = 5\):

4. **Final Equation**:
   \[
   \frac{(x + 5)^2}{5^2} + \frac{(y + 2)^2}{11^2} = 1
   \]
   Simplifying further:
   \[
   \frac{(x + 5)^2}{25} + \frac{(y + 2)^2}{121} = 1
   \]

### Answer:
The standard form of the equation is:
\[
\frac{(x + 5)^2}{25} + \frac{(y + 2)^2}{121} = 1
\]

Would you like a more detailed explanation of each step? 

Here are some related questions:
1. How would the equation change if the major axis was horizontal?
2. What is the equation if the center were at the origin?
3. How do we find the foci of this ellipse?
4. Can we determine the eccentricity of this ellipse?
5. What if the lengths of the axes were different? How would that affect \(a\) and \(b\)?

**Tip:** For ellipses, always check if the major axis is horizontal or vertical—it affects the position of \(a\) and \(b\) in the equation.

Find the standard form of the equation of the ellipse with major axis vertical, length 22, minor axis length 10, and center (-5, -2).

Find the standard form of the equation for an ellipse with a vertical major axis, length 22, minor axis length 10, and center at (-5, -2). This problem covers analytic geometry concepts, including ellipse equations and axis orientation.

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