key term - Horizontal Ellipse

Definition

A horizontal ellipse is a type of conic section that is defined as the set of all points in a plane where the sum of the distances from two fixed points, called the foci, is constant. Unlike a circle, which is a special case of an ellipse where the two foci coincide, an ellipse has two distinct foci that are not equidistant from the center.

5 Must Know Facts For Your Next Test

1. The equation of a horizontal 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 foci of a horizontal ellipse lie on the major axis, and the distance between them is $2c$, where $c = \sqrt{a^2 - b^2}$.
3. The eccentricity of a horizontal ellipse is calculated as $e = \frac{c}{a}$, where $c$ is the distance between the center and either focus.
4. The area of a horizontal ellipse is given by the formula $\pi ab$, where $a$ and $b$ are the lengths of the major and minor axes, respectively.
5. Horizontal ellipses can be used to model various real-world phenomena, such as the orbits of planets around the sun, the cross-sections of certain types of lenses, and the shapes of some architectural structures.

Review Questions

• Explain the relationship between the foci and the major axis of a horizontal ellipse.
  • The foci of a horizontal ellipse lie on the major axis, and the distance between them is $2c$, where $c = \sqrt{a^2 - b^2}$ and $a$ is the length of the major axis. This means that the foci are not equidistant from the center of the ellipse, and the sum of the distances from any point on the ellipse to the two foci is constant, which is a defining property of an ellipse.
• Describe how the eccentricity of a horizontal ellipse is calculated and what it represents.
  • The eccentricity of a horizontal ellipse is calculated as $e = \frac{c}{a}$, where $c$ is the distance between the center and either focus, and $a$ is the length of the major axis. Eccentricity is a measure of how elongated the ellipse is, with a value between 0 and 1. An eccentricity of 0 corresponds to a circle, while a value closer to 1 indicates a more elongated ellipse.
• Discuss how the area of a horizontal ellipse is calculated and the significance of this formula.
  • The area of a horizontal ellipse is given by the formula $\pi ab$, where $a$ and $b$ are the lengths of the major and minor axes, respectively. This formula is derived from the definition of an ellipse and the properties of circles. The area of a horizontal ellipse is an important characteristic that can be used to model and analyze various real-world phenomena, such as the cross-sections of certain types of lenses or the shapes of architectural structures.

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.

Eccentricity:

A measure of how elongated an ellipse is, calculated as the ratio of the distance between the foci to the length of the major axis.

Major Axis:

The longest diameter of an ellipse, passing through the center and the two foci.