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.