Ellipse

Definition

An ellipse is a set of all points in a plane such that the sum of their distances from two fixed points (foci) is constant. It has an oval shape and can be represented by its standard equation.

5 Must Know Facts For Your Next Test

1. The standard form of an ellipse's equation is $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ for horizontal ellipses or $\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1$ for vertical ellipses, where $(h,k)$ is the center.
2. In an ellipse, $a$ is the length of the semi-major axis and $b$ is the length of the semi-minor axis.
3. The distance between the foci is given by $2c$, where $c = \sqrt{a^2 - b^2}$.
4. When $a = b$, the ellipse becomes a circle.
5. The eccentricity ($e$) of an ellipse measures its deviation from being circular and is calculated as $e = \frac{c}{a}$.

Review Questions

• What are the coordinates of the foci for an ellipse centered at $(0,0)$ with axes lengths 8 and 6?
• Write down the equation for an ellipse with center $(3,-4)$, major axis length 10 along x-axis, and minor axis length 6 along y-axis.
• How do you find the eccentricity of an ellipse?

Related terms

Hyperbola: A set of all points in a plane where the difference of their distances from two fixed points (foci) is constant.

Parabola: A set of all points in a plane equidistant from a given point (focus) and a given line (directrix).

Conic Sections: Curves obtained by intersecting a cone with a plane: circles, ellipses, parabolas, and hyperbolas.