Summary: The Ellipse

Key Equations

Horizontal ellipse, center at origin [latex]\dfrac{{x}^{2}}{{a}^{2}}+\dfrac{{y}^{2}}{{b}^{2}}=1,\text{ }a>b[/latex]
Vertical ellipse, center at origin [latex]\dfrac{{x}^{2}}{{b}^{2}}+\dfrac{{y}^{2}}{{a}^{2}}=1,\text{ }a>b[/latex]
Horizontal ellipse, center [latex]\left(h,k\right)[/latex] [latex]\dfrac{{\left(x-h\right)}^{2}}{{a}^{2}}+\dfrac{{\left(y-k\right)}^{2}}{{b}^{2}}=1,\text{ }a>b[/latex]
Vertical ellipse, center [latex]\left(h,k\right)[/latex] [latex]\dfrac{{\left(x-h\right)}^{2}}{{b}^{2}}+\dfrac{{\left(y-k\right)}^{2}}{{a}^{2}}=1,\text{ }a>b[/latex]

Key Concepts

  • An ellipse is the set of all points [latex]\left(x,y\right)[/latex] in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci).
  • When given the coordinates of the foci and vertices of an ellipse, we can write the equation of the ellipse in standard form.
  • When given an equation for an ellipse centered at the origin in standard form, we can identify its vertices, co-vertices, foci, and the lengths and positions of the major and minor axes in order to graph the ellipse.
  • When given the equation for an ellipse centered at some point other than the origin, we can identify its key features and graph the ellipse.
  • Real-world situations can be modeled using the standard equations of ellipses and then evaluated to find key features, such as lengths of axes and distance between foci.

Glossary

center of an ellipse the midpoint of both the major and minor axes

conic section any shape resulting from the intersection of a right circular cone with a plane

ellipse the set of all points [latex]\left(x,y\right)[/latex] in a plane such that the sum of their distances from two fixed points is a constant

foci plural of focus

focus (of an ellipse) one of the two fixed points on the major axis of an ellipse such that the sum of the distances from these points to any point [latex]\left(x,y\right)[/latex] on the ellipse is a constant

major axis the longer of the two axes of an ellipse

minor axis the shorter of the two axes of an ellipse