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The equation you've provided represents an ellipse in the standard form:

\[
\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1
\]

For the given equation:

\[
\frac{(x + 2)^2}{16} + \frac{(y - 5)^2}{25} = 1
\]

Here are the parameters for this ellipse:

- \( h = -2 \) and \( k = 5 \): These are the coordinates of the center of the ellipse, so the center is at (-2, 5).
- \( a^2 = 16 \) and \( b^2 = 25 \): This indicates that \( a = 4 \) and \( b = 5 \), which are the semi-major and semi-minor axes, respectively.
- The major axis is along the x-direction because \( a^2 \) is smaller than \( b^2 \).

The ellipse is oriented horizontally because the larger denominator is under the \( y \)-term. This means the longer axis of the ellipse runs vertically (along the y-axis).

Would you like further details on this topic or any help with another question? Here are a few things you might consider asking:

1. How can I find the vertices of this ellipse?
2. What are the foci of the ellipse?
3. How can I graph this ellipse?
4. Can you explain how the orientation of an ellipse is determined?
5. Could you derive the equation for the ellipse's directrices?

**Tip:** When graphing an ellipse, always start by locating the center, then measure out the distances of the axes from the center to get accurate positioning of the vertices and foci.

Solve the equation (x + 2)^2 / 16 + (y - 5)^2 / 25 = 1 for the ellipse.

Explore the properties of the ellipse given by the equation (x + 2)^2 / 16 + (y - 5)^2 / 25 = 1. Learn how to determine the center, axes lengths, and orientation of the ellipse. Suitable for students in Grades 10-12 studying analytic geometry.

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