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?