For lots more information on ellipses, see these earlier sections:
Summary:
Equations of ellipses
with centers at the origin
and foci on the $x$axis or $y$axis
In both cases:
$\,a > b > 0\,$
The length of the major axis (which contains the foci) is $\,2a\,$.
The length of the minor axis is $\,2b\,$.
The foci are determined by solving the equation $\,c^2 = a^2  b^2\,$ for $\,c\,$.
Foci on the $x$axis
Equation of Ellipse:
$$\frac{x^2}{\underset{\text{bigger}}{\underset{\uparrow}{a^2}}} + \frac{y^2}{b^2} = 1$$
When the foci are on the $\color{red}{x}$axis,
the bigger number ($\,a^2 > b^2\,$) is beneath the $\,\color{red}{x}^2\,$.
Coordinates of foci: $\,(c,0)\,$ and $\,(c,0)\,$

Foci on the $y$axis
Equation of Ellipse:
$$\frac{x^2}{b^2} + \frac{y^2}{\underset{\text{bigger}}{\underset{\uparrow}{a^2}}} = 1$$
When the foci are on the $\color{red}{y}$axis,
the bigger number ($\,a^2 > b^2\,$) is beneath the $\,\color{red}{y}^2\,$.
Coordinates of foci: $\,(0,c)\,$ and $\,(0,c)\,$

TIPS:

The key to recognizing the equation of an ellipse with center at the origin and foci on either the $x$axis or $y$axis is this:
 It has only $\,x^2\,$, $\,y^2\,$, and constant terms.

When the $\,x^2\,$ and $\,y^2\,$ terms are on the same side of the equation,
then they must have the same sign.

When the bigger number is beneath the
$\,x^2\,$ term, then the foci are on the $\,x\,$axis.
When the bigger number is beneath the
$\,y^2\,$ term, then the foci are on the $\,y\,$axis.
Look for the bigger denominator to tell you where the foci lie!

Cut down on memorization—just use standard techniques to get the $x$ and $y$ intercepts.
For example, in the equation $\displaystyle\,\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\,$:
Let $\,y = 0\,$ to get the $x$intercepts $\,a\,$ and $\,a\,$:
$\displaystyle\,\frac{x^2}{a^2} = 1 \ \Rightarrow\ x^2 = a^2 \ \Rightarrow\ x = \pm a\,$.
Let $\,x = 0\,$ to get the $y$intercepts $\,b\,$ and $\,b\,$:
$\displaystyle\,\frac{y^2}{b^2} = 1 \ \Rightarrow\ y^2 = b^2 \ \Rightarrow\ y = \pm b\,$.

To find the foci: $\,c^2\,$ is always the bigger denominator minus the smaller denominator.
$$c^2 = \text{bigger denominator}  \text{smaller denominator}$$
EXAMPLE: Graphing an Ellipse
Graph: $100  4x^2 = 25y^2$
Initial thoughts:
There are only $\,x^2\,$, $\,y^2\,$ and constant terms.
When the $\,x^2\,$ and $\,y^2\,$ terms are on the same side of the equation, they have the same sign.
It's an ellipse with center at the origin and foci on either the $x$axis or $y$axis!
Solution:
$$
\begin{gather}
100  4x^2 = 25y^2\cr\cr
4x^2 + 25y^2 = 100\cr\cr
\frac{4x^2}{100} + \frac{25y^2}{100} = 1\cr\cr
\frac{4x^2}{100}\cdot\frac{\frac 14}{\frac 14} + \frac{25y^2}{100}\cdot\frac{\frac 1{25}}{\frac 1{25}} = 1\cr\cr
\frac{x^2}{25} + \frac{y^2}{4} = 1\cr\cr
\end{gather}
$$
$x$intercepts (set $\,y = 0\,$): $\,x = \pm 5\,$
$y$intercepts (set $\,x = 0\,$): $\,y = \pm 2\,$
Foci:
The bigger number is under the $\,x^2\,$, so the foci are on the $x$axis.
$\,c^2 = 25  4 = 21\,$
$\,c = \pm\sqrt{21}$
$\,c\approx \pm 4.6\,$


EXAMPLE: Finding the Equation of an Ellipse
Find the equation of the following ellipse:
 center at the origin
 major axis along the $y$axis, with length $\,8\,$
 minor axis has length $\,6\,$
Also, find the coordinates of the foci.
Solution:

Since the major axis is along the $y$axis, the form of the equation is
$$
\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1
$$

$\,a\,$ is half the length of the major axis, which is $\,\frac{8}{2} = 4\,$

$\,b\,$ is half the length of the minor axis, which is $\,\frac{6}{2} = 3\,$

The equation is:
$$
\begin{gather}
\frac{x^2}{3^2} + \frac{y^2}{4^2} = 1\cr\cr
\frac{x^2}9 + \frac{y^2}{16} = 1
\end{gather}
$$

Foci:
$\,c^2 = a^2  b^2 = 16  9 = 7\,$, so $\,c = \pm\sqrt{7}\,$
Coordinates of foci: $\,(0,\sqrt{7})\,$ and $\,(0,\sqrt{7})\,$
Master the ideas from this section
by practicing the exercise at the bottom of this page.
When you're done practicing, move on to:
Definition of a Hyperbola