Ellipse

Definition

Similar to a parabola, an ellipse is defined by some fixed distance from a pair of points, called foci. The ellipse is the set of points whose sum of distances from the foci is fixed.

A more explicit description, starts by fixing the foci, f and g, and fixing some distance, r, greater than or equal to the distance between the foci. To check if a point v with coordinates (x, y) is on the ellipse, we take the distance from v to f and add it to the distance from v to g. If the resulting number is r, then v is on the ellipse.

It is not obvious from the definition, but an ellipse forms an oval, or circular, shape. The line defined by the foci is the major axis, and intersects the ellipse in two points, called vertices. The line that is perpendicular to the major axis is the minor axis. In the precaculus courses at UC Riverside we focus on when the major axis is a horizontal or vertical line.

Algebraic Expression

Very similarly to how the equation of a parabola is centered around the vertex, the equation for an ellipse is centered around a point called the center, which happens to be the midpoint of the line segment between the foci.

Given the center of the ellipse is (0, 0), with foci (-c, 0) and (c, 0) and vertices (a, 0) and (-a, 0), the equation for the ellipse is

${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1~{\text{ where }}a>b>0,~{\text{ and }}b^{2}=a^{2}-c^{2}$

This equation gives us an ellipse that is wider than it is tall. If we want an ellipse that is taller than it is wide we change the foci to (0, c) and (0, -c), the vertices to (0, a) an (0, -a), and switch the positions of the x's and y's in the equation.

We can also define an ellipse with center not at (0,0), but instead at (h, k) by replacing x with (x - h) and y with (y - k).

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