📈college algebra review

key term - Ellipses

Definition

An ellipse is a closed, two-dimensional 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 $a$ and $b$ are the lengths of the major and minor axes, respectively.
  2. Ellipses can be used to model various real-world phenomena, such as the orbits of planets around the sun, the shape of certain architectural structures, and the cross-section of an egg.
  3. Parametric equations can be used to represent the shape of an ellipse, where the $x$ and $y$ coordinates are expressed in terms of a parameter $t$.
  4. 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 eccentricity of the ellipse.
  5. Ellipses have several important properties, such as the fact that the sum of the distances from any point on the ellipse to the two foci is constant, and that the tangent line to the ellipse at any point is perpendicular to the line connecting that point to one of the foci.

Review Questions

  • Explain how parametric equations can be used to represent the shape of an ellipse.
    • Parametric equations allow for the representation of an ellipse by expressing the $x$ and $y$ coordinates in terms of a parameter $t$. Typically, the parametric equations for an ellipse centered at the origin with major axis $2a$ and minor axis $2b$ are given by $x(t) = a\cos(t)$ and $y(t) = b\sin(t)$, where $t$ is a parameter that varies from $0$ to $2\pi$. This parametric representation enables the description of the ellipse's shape and allows for the exploration of its properties, such as the relationship between the foci and the points on the curve.
  • Describe the key properties of an ellipse and how they relate to its representation using parametric equations.
    • An ellipse has several important properties that are reflected in its parametric equation representation. Firstly, the sum of the distances from any point on the ellipse to the two foci is constant, which is a fundamental property of the shape. This can be seen in the parametric equations, where the $x$ and $y$ coordinates are determined by the major and minor axis lengths, $a$ and $b$, respectively. Additionally, the tangent line to the ellipse at any point is perpendicular to the line connecting that point to one of the foci. This property can be derived from the parametric equations by examining the derivative of the $x$ and $y$ functions with respect to the parameter $t$.
  • Analyze how the parameters $a$ and $b$ in the standard form equation of an ellipse, $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, influence the shape and orientation of the ellipse, and discuss the relationship between these parameters and the parametric equation representation.
    • The parameters $a$ and $b$ in the standard form equation of an ellipse, $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, directly influence the shape and orientation of the ellipse. The value of $a$ determines the length of the major axis, while $b$ determines the length of the minor axis. When $a = b$, the ellipse becomes a circle. As the ratio of $a$ to $b$ increases, the ellipse becomes more elongated. This relationship between the major and minor axis lengths is also reflected in the parametric equation representation, where $a$ and $b$ appear as the coefficients of the $\cos(t)$ and $\sin(t)$ terms, respectively. The parametric equations allow for a more dynamic visualization of how changes in $a$ and $b$ affect the shape and size of the ellipse.

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