Ellipse

Definition

An ellipse is a closed, two-dimensional geometric shape that resembles an elongated circle. It is defined as the set of all points on a plane where the sum of the distances from two fixed points, called the foci, is constant.

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 eccentricity of an ellipse is calculated 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.
3. Ellipses have two foci, which are two fixed points on the major axis that determine the shape of the ellipse.
4. The sum of the distances from any point on the ellipse to the two foci is constant and equal to the length of the major axis.
5. Ellipses are used to model various natural and man-made phenomena, such as the orbits of planets, the shape of a football, and the cross-section of a cylindrical pipe.

Review Questions

• Explain the relationship between the major axis, minor axis, and eccentricity of an ellipse.
  • The major axis and minor axis of an ellipse are the longest and shortest diameters of the shape, respectively. The eccentricity is a measure of how much the ellipse deviates from a perfect circle, ranging from 0 (a circle) to 1 (a line segment). The eccentricity is calculated 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. As the eccentricity increases, the ellipse becomes more elongated, with the major axis becoming significantly longer than the minor axis.
• Describe how the equation of an ellipse in standard form can be used to determine the properties of the shape.
  • The standard form equation of an ellipse 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. From this equation, we can determine several key properties of the ellipse. The lengths of the major and minor axes are directly given by the values of $a$ and $b$, respectively. The eccentricity can be calculated using the formula $e = \sqrt{1 - \frac{b^2}{a^2}}$. Additionally, the location of the foci can be found using the formula $c = \sqrt{a^2 - b^2}$, where $c$ is the distance from the center of the ellipse to each focus.
• Explain the significance of the foci in an ellipse and how they relate to the properties of the shape.
  • The foci of an ellipse are two fixed points on the major axis that determine the shape of the ellipse. The sum of the distances from any point on the ellipse to the two foci is constant and equal to the length of the major axis. This property is fundamental to the definition of an ellipse and has important implications. The location of the foci, as determined by the eccentricity, affects the overall shape of the ellipse. As the eccentricity increases, the foci move farther apart, and the ellipse becomes more elongated. The foci also play a crucial role in various applications of ellipses, such as in the design of reflectors and lenses, where the focal points are used to concentrate or disperse light or other forms of energy.

Related terms

Eccentricity: A measure of how much an ellipse deviates from a perfect circle, ranging from 0 (a circle) to 1 (a line segment).

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

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