All about Ellipses

An ellipse is defined as the set of all points where the product of the distances from two fixed points is constant.

The foci of an ellipse must lie on the minor axis.

The center of an ellipse is the intersection of the major and minor axes.

The length of the major axis of an ellipse is denoted by the value $a$.

The co-vertices of an ellipse are the endpoints of the major axis.

In an ellipse, the distance from the center to a focus is denoted as $c$, and $c^2 = a^2 + b^2$.

In the equation of an ellipse in general form, $Ax^2 + Cy^2 + Dx + Ey + F = 0$, the coefficients $A$ and $C$ must have opposite signs.

If $a = b$ in the standard equation of an ellipse, the ellipse is actually a circle.

In the standard form equation of an ellipse with a horizontal major axis, the larger denominator is under the $y$ term.

Halley's Comet follows an elliptical orbit around the Earth.

The vertices of the ellipse are located at $(h \pm a, k)$ when the major axis is vertical.

The foci of an ellipse are defined as the points where the minor axis intersects ellipse.

The distance from the center of an ellipse to a co-vertex is always greater than distance from the center to a vertex.

If the equation of an ellipse is $\frac{(x-2)^2}{9} + \frac{(y+1)^2}{4} = 1$, the center of the ellipse is at $(2, -1)$.

In an ellipse, the major axis is always longer than the minor axis.

The equation $\frac{x^2}{16} + \frac{y^2}{25} = 1$ represents an ellipse with a horizontal major axis.

The standard form of an ellipse with a vertical major axis and center at the origin is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where $a > b$.

The whispering gallery effect is experienced in circular rooms.

Given an ellipse with equation $\frac{(x+3)^2}{4} + \frac{(y-2)^2}{25}=1$, the length of the minor axis is 4.

A key step in converting an ellipse equation from standard to general form is to eliminate the fractions.

If the equation of an ellipse is $\frac{x^2}{4} + \frac{y^2}{9} = 1$, the coordinates of the co-vertices are $(\pm 3, 0)$.

If $a = 5$ and $b = 3$ for an ellipse, then $c = 4$, where $c$ is the distance from the center to each focus.

The equation $x^2 + 4y^2 + 6x - 8y + 9 = 0$ represents an ellipse.

The minor axis of an ellipse always passes through the foci.

Given the equation $\frac{(x-1)^2}{4} + \frac{(y+2)^2}{9} = 1$, the major axis has a length of 3.

In an ellipse, the sum of the distances from any point on the ellipse to the two foci is always equal to $2a$, where $a$ is the distance from the center to a vertex.

If the center of an ellipse is at $(0,0)$, and a focus is at $(5,0)$, then the other focus must be at $(-5, 0)$.

The vertices of an ellipse are the intersection points of ellipse with its major axis.

The line segment connecting the co-vertices is the major axis.

Given that the vertices of an ellipse are at $(-4, 0)$ and $(4, 0)$, the center of the ellipse is at the origin.

The equation of an ellipse with center at $(0,0)$, major axis of length $6$ along the y-axis, and minor axis of length $4$ along the x-axis is $\frac{x^2}{9} + \frac{y^2}{4} = 1$.

The standard form of an ellipse centered at $(h, k)$ can always be written such that $a > b$.

For an ellipse, if $a = 4$ and $c = 5$, then $b = 3$.

If vertices of an ellipse are at $(0, \pm 6)$ and foci are at $(0, \pm 4)$, the equation of the ellipse is $\frac{x^2}{20} + \frac{y^2}{36} = 1$.