Ellipse

Definition

An ellipse is a closed, two-dimensional shape that is defined by the set of all points on a plane where the sum of the distances from two fixed points, called the foci, is constant. Ellipses are a fundamental concept in mathematics, physics, and engineering, with applications in various fields, including optics, astronomy, and structural design.

5 Must Know Facts For Your Next Test

1. An ellipse can be represented by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $a$ and $b$ are the lengths of the major and minor axes, respectively.
2. The major axis of an ellipse is the longest diameter, and the minor axis is the shortest diameter. The lengths of these axes determine the shape and size of the ellipse.
3. Ellipses can be rotated in the coordinate plane, and the equation of the rotated ellipse can be expressed in terms of the angle of rotation.
4. In polar coordinates, an ellipse can be represented by the equation $r = \frac{ab}{\sqrt{b^2 \cos^2 \theta + a^2 \sin^2 \theta}}$, where $a$ and $b$ are the lengths of the major and minor axes, and $\theta$ is the angle from the positive $x$-axis.
5. Ellipses are used to model various natural and man-made phenomena, such as the orbits of planets around the sun, the shape of a football, and the cross-section of an architectural dome.

Review Questions

• Explain how the equation of an ellipse, $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, relates to the lengths of the major and minor axes.
  • The equation of an ellipse, $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, directly relates to the lengths of the major and minor axes. The parameter $a$ represents the length of the major axis, and the parameter $b$ represents the length of the minor axis. The shape and size of the ellipse are determined by the values of $a$ and $b$, with the major axis being the longest diameter and the minor axis being the shortest diameter of the ellipse.
• Describe how the concept of an ellipse is connected to the topic of systems of nonlinear equations and inequalities in two variables.
  • Ellipses are a type of conic section, which can be represented by a system of nonlinear equations or inequalities in two variables. The equation of an ellipse, $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, is a nonlinear equation that defines the shape of the ellipse. When combined with other equations or inequalities, such as those representing constraints or boundaries, the system of nonlinear equations and inequalities can be used to model and analyze various real-world problems involving ellipses, such as optimization problems, resource allocation, and decision-making.
• Analyze how the rotation of axes can affect the equation and properties of an ellipse, and explain the significance of this concept in the context of conic sections.
  • The rotation of axes can significantly impact the equation and properties of an ellipse. When an ellipse is rotated in the coordinate plane, the equation of the ellipse can be expressed in terms of the angle of rotation, $\theta$. This transformed equation, which involves trigonometric functions, allows for the analysis of the ellipse's orientation, eccentricity, and other characteristics. The concept of rotating axes is important in the study of conic sections because it enables the classification and analysis of different types of conic sections, such as circles, parabolas, and hyperbolas, which can be obtained by varying the angle of rotation and the parameters of the ellipse. Understanding the effects of axis rotation is crucial for solving problems involving the transformation and analysis of conic sections in various applications, including physics, engineering, and computer graphics.

Related terms

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

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

Conic Sections: The shapes formed by the intersection of a plane and a cone, including the ellipse, circle, parabola, and hyperbola.