CSPE - Ellipses Lesson

Ellipses

Next up in our study of conic sections, we will discuss ellipses. An ellipse is the set of points in a plane such that the sum of the distances from two fixed points, called the foci, remains constant.

image of a plane slicing a cone to form an ellipse

Watch the video to see an animation of how the sum of the distances from two fixed points, called the foci, remains constant in an ellipse.

The major axis of an ellipse is the distance between the vertices. The major axis can be horizontal or vertical.

Standard Form Equations for Ellipses

$LaTeX: \frac{\left(x-h\right)^2}{a^2}+\frac{\left(y-k\right)^2}{b^2}=1$ $\frac{\left(x-h\right)^2}{a^2}+\frac{\left(y-k\right)^2}{b^2}=1$	$LaTeX: \frac{\left(x-h\right)^2}{b^2}+\frac{\left(y-k\right)^2}{a^2}=1$ $\frac{\left(x-h\right)^2}{b^2}+\frac{\left(y-k\right)^2}{a^2}=1$

Opens Horizontally	Opens Vertically
Center: (h, k)	Center: (h, k)
Vertices: $LaTeX: \left(h\pm a,\:k\right)$ $\left(h\pm a,\:k\right)$	Vertices: $LaTeX: \left(h,\:k\pm a\right)$ $\left(h,\:k\pm a\right)$
Co-Vertices: $LaTeX: \left(h,\:k\pm b\right)$ $\left(h,\:k\pm b\right)$	Co-Vertices: $LaTeX: \left(h\pm b,\:k\right)$ $\left(h\pm b,\:k\right)$
Foci: $LaTeX: \left(h\pm c,\:k\right)$ $\left(h\pm c,\:k\right)$	Foci: $LaTeX: \left(h,\:k\pm c\right)$ $\left(h,\:k\pm c\right)$
The equation for finding c: $c^2=a^2-b^2$

Let's try writing the equation given the graph of an ellipse.

Watch this video to practice converting from general form to standard form.

Properties of Ellipses Practice

Given the equation of each ellipse, find the correct properties. Then sketch the graph on your own paper with the requested information.

IMAGES SOURCE: SAYLORDOTORG, CREATED BY GAVS