Major and minor axes

Definition

The major axis of an ellipse is the longest diameter, passing through the center and both foci. The minor axis is perpendicular to the major axis at the center, representing the shortest diameter.

5 Must Know Facts For Your Next Test

1. The lengths of the major and minor axes are denoted as $2a$ and $2b$ respectively, where $a > b$.
2. In standard form, an ellipse centered at $(0,0)$ with a horizontal major axis has the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.
3. If $a = b$, the ellipse becomes a circle.
4. The endpoints of the major axis are called vertices, located at $(\pm a, 0)$ for a horizontally oriented ellipse.
5. The minor axis intersects the major axis at the center of the ellipse.

Review Questions

• What are the lengths of the major and minor axes in terms of $a$ and $b$?
• Where are the vertices located if an ellipse is centered at $(0,0)$ with a horizontal major axis?
• How does an ellipse change when $a = b$?

Related terms

Foci: Two fixed points on the interior of an ellipse used in its formal definition. The sum of distances from any point on the ellipse to each focus is constant.

Center: The midpoint between the foci where both axes intersect. In equations, it is typically represented as $(h,k)$.

$eccentricity (e)$: $e$ describes how elongated an ellipse is. Calculated as $e = \sqrt{1 - \frac{b^2}{a^2}}$, where $0 < e < 1$ for ellipses.