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Geometry - The Ellipse

An ellipse is defined as the set of all points where the sum of the distances from two fixed points, called foci, is constant. The line through the foci intersects the ellipse at two points called vertices. The major axis connects the vertices and its midpoint is the center of the ellipse. The standard equation of an ellipse expresses the relationship between the distances to the foci.

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24 views2 pages

Geometry - The Ellipse

An ellipse is defined as the set of all points where the sum of the distances from two fixed points, called foci, is constant. The line through the foci intersects the ellipse at two points called vertices. The major axis connects the vertices and its midpoint is the center of the ellipse. The standard equation of an ellipse expresses the relationship between the distances to the foci.

Uploaded by

Jason Costanzo
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© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Notes – Ellipse

The ellipse can also be defined as a locus (collection) of points satisfying a geometric property.

Definition of Ellipse. An ellipse is the set of all points (𝑥, 𝑦) in a plane, the sum of whose distances from
two distinct fixed points (foci) is constant.

The line through the foci intersects the ellipse at two points called vertices. The chord joining the
vertices is the major axis, and its midpoint is the center of the ellipse. The chord perpendicular to the
major axis at the center is the minor axis of the ellipse.

𝑑1 + 𝑑2 is constant

Click here to join me in my woodshop to visualize the definition of an ellipse.

Click here to see the development of the standard form of the equation of an ellipse with center, (ℎ, 𝑘);
vertices, (ℎ ± 𝑎, 𝑘); and foci, (ℎ ± 𝑐, 𝑘).

Standard Equation of an Ellipse. The standard form of the equation of an ellipse, with center (ℎ, 𝑘) and
major and minor axes of lengths 2𝑎 and 2𝑏, respectively, where 𝑎 > 𝑏 > 0, is

(𝑥 − ℎ)2 (𝑦 − 𝑘)2
+ =1 (𝑚𝑎𝑗𝑜𝑟 𝑎𝑥𝑖𝑠 𝑖𝑠 ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙)
𝑎2 𝑏2

(𝑥 − ℎ)2 (𝑦 − 𝑘)2
+ =1 (𝑚𝑎𝑗𝑜𝑟 𝑎𝑥𝑖𝑠 𝑖𝑠 𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙)
𝑏2 𝑎2

The foci lie on the major axis, 𝑐 units from the center, with 𝑐 2 = 𝑎2 − 𝑏 2. If the center is at the origin
(0, 0), the equation takes one of the following forms.

𝑥2 𝑦2
+ =1 (𝑚𝑎𝑗𝑜𝑟 𝑎𝑥𝑖𝑠 𝑖𝑠 ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙)
𝑎2 𝑏 2

𝑥2 𝑦2
+ =1 (𝑚𝑎𝑗𝑜𝑟 𝑎𝑥𝑖𝑠 𝑖𝑠 𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙)
𝑏 2 𝑎2
The eccentricity 𝑒 of an ellipse is given by the ratio
𝑐
𝑒= , 0<𝑒<1
𝑎

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