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Equations of Ellipses and Hyperbolas

Videos and lessons with examples and solutions to help High School students derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.

Common Core: HSG-GPE.A.3

Deriving the equation of an ellipse
Deriving the equation of an ellipse from the property of each point being the same total distance from the two foci. Used as an example of manipulating equations with square roots.

Derivation of ellipse formula
We use coordinate proofs to develop the standard equation of a ellipse. We begin by defining the ellipse as a locus of points. Make comparisons between horizontal and vertical ellipses.

Graph and Write Equations of Ellipses
A discussion on the components and equations of ellipses
An ellipse has two foci, a major axis, a minor axis, a center, vertices and co-vertices.

Hyperbola - Definition and derivation of the equation
This video discusses what hyperbolas are and derive the equation for a hyperbola based on it's definition which is the difference in distances from a point on the curve to 2 fixed focal points. This derivation is long and very algebra intensive but the end result is a simple equation to encompass a hyperbola.

Proof of the hyperbola foci formula.

Locus of Points Definition of an Ellipse, Hyperbola, Parabola, and Oval of Cassini
Given two points, f1 and f2 (the foci), an ellipse is the locus of points P  such that the sum of the distances from P to f1 and to f2 is a constant.
A hyperbola is the locus of points P such that the absolute value of the difference between the distances from P to f1 and to f2 is a constant.
An oval of Cassini is the locus of points P such that the product of the distances from P to f1 and to f2 is a constant (b2 here).
A parabola is the locus of points P such that the distance from P to a point f (the focus) is equal to the distance from P to a line L  (the directrix).
This Demonstration illustrates those definitions by letting you move a point along the figure and watch the relevant distances change.

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