Ellipse

Geometrical definition

Given two points $F_{1}$ and $F_{2}$ of the plane, and a real number $2a$ greater than the distance $F_{1}F_{2}$ , the ellipse with foci $F_{1},F_{2}$ and major axis $2a$ is the set of points $P$ of the plane such that:
$PF_{1}+PF_{2}=2a.$

Explanation

This definition is called the gardener’s method: plant two stakes at $F_{1}$ and $F_{2}$ , loop a string around them, and trace the curve keeping the string taut. This will produce an ellipse. This construction generalizes the one for the circle, where a single stake and a taut string of fixed length give the set of points at constant distance from a single center.

Cartesian equation definition

An ellipse is the set of points $M:=(x,y)\in\R^2$ satisfying an equation
$\left(\frac{x-x_{0}}{a}\right)^{2} + \left(\frac{y-y_{0}}{b}\right)^{2} = 1,$ where $a,b$ are positive real numbers called respectively "semi major-axis" and "semi minor-axis". Also, $(x_0,y_0)$ is the center of the ellipse.

Examples

Let $C$ be the (double) cone in $\R^3$ with vertex at $0$ obtained by rotation the line generated by $(1,0,1)$ around the $z$ -axis. Then the plane section of a cone is an ellipse, when the plane doesn’t pass through $0$ and has the angle with the horizontal plane less than $\frac{\pi}{4}$ .

Practice this concept with exercises

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Which of the following equations define an ellipse ?
1. $x^{2}+2y^{2}=-1$
2. $x^2+y^2=2$
3. $4x^{2}+9y^{2}=36$
4. $5x^{2}+2y^{2}=0$
5. $x^{2}-y^{2}=1$
6. $x^{2}+2x+y^{2}=1$
7. Show that any circle is, in particular, an ellipse.
Open exerciseDifficulty 26/100 · 1 solution · 0 hints

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Marden’s theorem

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