Ellipse

Definition

An ellipse is a closed, two-dimensional shape that is the set of all points on a plane where the sum of the distances from two fixed points, called the foci, is constant. It is one of the fundamental conic sections, along with the circle, parabola, and hyperbola.

5 Must Know Facts For Your Next Test

1. The equation of an ellipse in standard form is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $a$ is the length of the major axis and $b$ is the length of the minor axis.
2. The distance between the foci of an ellipse is $2c$, where $c = \sqrt{a^2 - b^2}$.
3. The eccentricity of an ellipse, denoted by $e$, is a measure of how elongated the ellipse is. It is defined as $e = \sqrt{1 - \frac{b^2}{a^2}}$, where $0 < e < 1$.
4. Ellipses have many real-world applications, such as in the design of bridges, the paths of planetary orbits, and the shapes of reflectors and lenses.
5. The area of an ellipse is given by the formula $\pi ab$, where $a$ is the length of the major axis and $b$ is the length of the minor axis.

Review Questions

• Explain the relationship between the foci and the major and minor axes of an ellipse.
  • The foci of an ellipse are the two fixed points that define the shape of the ellipse. The major axis of an ellipse is the longest diameter, passing through the foci, while the minor axis is the shortest diameter, perpendicular to the major axis. The distance between the foci is determined by the lengths of the major and minor axes, specifically $2c = \sqrt{a^2 - b^2}$, where $a$ is the length of the major axis and $b$ is the length of the minor axis.
• Describe how the eccentricity of an ellipse is related to its shape and the lengths of the major and minor axes.
  • The eccentricity of an ellipse, denoted by $e$, is a measure of how elongated the ellipse is. It is defined as $e = \sqrt{1 - \frac{b^2}{a^2}}$, where $a$ is the length of the major axis and $b$ is the length of the minor axis. When $e = 0$, the ellipse becomes a circle, and as $e$ approaches 1, the ellipse becomes more elongated. The eccentricity is directly related to the ratio of the major and minor axis lengths, with a higher eccentricity indicating a more elongated ellipse.
• Explain the real-world applications of ellipses and how their properties, such as the major and minor axes and eccentricity, are utilized in these applications.
  • Ellipses have numerous real-world applications, including in the design of bridges, the paths of planetary orbits, and the shapes of reflectors and lenses. The properties of an ellipse, such as the lengths of the major and minor axes and its eccentricity, are crucial in these applications. For example, the shape of a bridge's arch is often designed as an ellipse to provide structural stability and aesthetic appeal. The elliptical orbits of planets around the Sun are a consequence of the inverse-square law of gravitation. Additionally, the eccentricity of an ellipse determines the shape of reflectors and lenses, which is important in the design of optical devices like telescopes and cameras.

Related terms

Foci: The two fixed points that define an ellipse, where the sum of the distances from any point on the ellipse to the foci is constant.

Major Axis: The longest diameter of an ellipse, passing through the foci.

Minor Axis: The shortest diameter of an ellipse, perpendicular to the major axis.