An ellipse is a set of all points in a plane where the sum of the distances from two fixed points (foci) is constant. It is an important type of conic section.
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The standard form equation of an ellipse centered at the origin: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$
In the standard form, if $a > b$, the major axis is along the x-axis; if $b > a$, it is along the y-axis.
The distance between the center and each focus is given by $c = \sqrt{a^2 - b^2}$ for horizontal ellipses and $c = \sqrt{b^2 - a^2}$ for vertical ellipses.
The eccentricity of an ellipse, denoted as $e$, measures its deviation from being circular and is calculated as $e = \frac{c}{a}$ for horizontal ellipses or $e = \frac{c}{b}$ for vertical ellipses.
An ellipse has two axes of symmetry: the major axis (longest diameter) and minor axis (shortest diameter).
Review Questions
What is the standard form equation of an ellipse centered at $(h, k)$?
How do you find the foci of an ellipse given its equation in standard form?
What distinguishes an ellipse from other conic sections like parabolas and hyperbolas?
Related terms
Conic Section: A curve obtained by intersecting a cone with a plane; includes circles, ellipses, parabolas, and hyperbolas.
Major Axis: The longest diameter of an ellipse, passing through both foci.
Eccentricity: A measure of how much an ellipse deviates from being circular, calculated as the ratio of the distance between foci to the length of the major axis.