Summary: Equations of Ellipses in Standard Form

Summary: Equations of Ellipses in Standard Form

 

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:

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:

Also, find the coordinates of the foci.

Solution:
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


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