The endpoints of the major axis are critical when dealing with ellipses. An ellipse is defined by two main axes: the **major axis** and the **minor axis**. The **major axis** is the longest diameter of the ellipse, and it passes through the **center** and both **foci** of the ellipse.

For an ellipse centered at \((h, k)\) with a **horizontal major axis**, its equation is:

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

The **endpoints of the major axis** for this form are:

\[
(h - a, k) \quad \text{and} \quad (h + a, k)
\]

If the ellipse is centered at \((h, k)\) with a **vertical major axis**, the equation becomes:

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

In this case, the **endpoints of the major axis** are:

\[
(h, k - a) \quad \text{and} \quad (h, k + a)
\]

Here, \(a\) represents the length from the center to each endpoint of the major axis, and \(2a\) is the length of the major axis.

Would you like more details about ellipses or any other specific aspect?

1. What is the difference between the major and minor axes of an ellipse?
2. How do you find the foci of an ellipse?
3. How can you determine whether an ellipse is horizontal or vertical?
4. What is the relationship between the major axis and the eccentricity of an ellipse?
5. How can you derive the equation of an ellipse given its endpoints?

**Tip:** For any ellipse, remember that the larger value between \(a\) and \(b\) determines which axis is the major axis.

Understand how to find the endpoints of the major axis of an ellipse. This guide explores ellipses, focusing on the major and minor axes, key formulas, and the relationship between the center and foci. Ideal for students in Grades 10-12.

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