Equations of Ellipses in Standard Form:
Foci on the $y$-axis

LESSON READ-THROUGH
by Dr. Carol JVF Burns (website creator)
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In an earlier lesson, the equation of an ellipse with center at the origin and foci on the $x$-axis was derived, in great detail.

You may want to review this earlier lesson before studying the ‘in-a-nutshell’ derivation here.

Derivation of Ellipse Equation:
Center at the Origin, Foci on the $y$-axis

Although it is much shorter, this derivation should look strikingly familiar to the earlier derivation.

Notice also that the variables $\,a\,$, $\,b\,$ and $\,c\,$ have the same meaning as in the earlier derivation,
and the relationship between these three variables is the same.

  • Position the ellipse:
    As shown at right, position an ellipse with center at the origin
    and foci (marked with ‘x’) on the $y$-axis.
  • Notation ($\,c\,$ and $\,a\,$):
    Define:
    • $c = \,$ the distance from the center (the origin) to each focus.
      Since $\,c\,$ is a distance, $\,c > 0\,$.
      The coordinates of the foci are therefore $\,(0,c)\,$ and $\,(0,-c)\,$.
    • $a = \,$ the distance from the center to each vertex.
      Since $\,a\,$ is a distance, $\,a > 0\,$.
      The coordinates of the vertices are therefore $\,(0,a)\,$ and $\,(0,-a)\,$.
  • Ellipse constant:
    The ellipse constant is the length of the major axis, which is $\,2a\,$.
  • Use the definition of ellipse on a typical point:
    Let $\,(x,y)\,$ be a typical point on the ellipse,
    so that the sum of its distances to the foci is the ellipse constant ($\,2a\,$).

    That is, $$ \cssId{s22}{\biggl( \text{distance from } (x,y) \text{ to } (0,-c) \biggr) + \biggl( \text{distance from } (x,y) \text{ to } (0,c) \biggr) = 2a} $$ Use the distance formula and make obvious simplications: $$ \begin{gather} \cssId{s24}{\sqrt{\bigl(x - 0\bigr)^2 + \bigl(y-(-c)\bigr)^2} + \sqrt{\bigl(x - 0\bigr)^2 + \bigl(y-c\bigr)^2} = 2a}\cr\cr \cssId{s25}{\sqrt{x^2 + (y+c)^2} + \sqrt{x^2 + (y-c)^2} = 2a} \end{gather} $$
  • Define $\,b\,$; get relationship between $\,a\,$, $\,b\,$ and $\,c\,$:
    As before, define $\,b\,$ as the distance from the center to an endpoint of the minor axis.
    Then, $\,c^2 + b^2 = a^2\,$.
    Equivalently, $\,b^2 = a^2 - c^2\,$ and $\,c^2 = a^2 - b^2\,$.
    Again, $\,a > b\,$.
  • Rewrite equation to eliminate $\,c\,$, and simplify:
    $$ \begin{gather} \cssId{s53}{a^2x^2 + b^2y^2 = a^2b^2}\cr\cr \cssId{s54}{\frac{a^2x^2}{a^2b^2} + \frac{b^2y^2}{a^2b^2} = \frac{a^2b^2}{a^2b^2}}\cr\cr \cssId{s55}{\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1} \end{gather} $$
Summary:
The equation of an ellipse with center at the origin and foci along the $\,y$-axis is $$ \cssId{s58}{\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1} $$ where:
  • $a > b > 0$
  • The length of the major axis (which lies on the $\,y$-axis) is $\,2a\,$.
  • The length of the minor axis (which lies on the $\,x$-axis) is $\,2b\,$.
  • The foci are determined by solving the equation $\,c^2 = a^2 - b^2\,$ for $\,c\,$.
    The coordinates of the foci are $\,(0,-c)\,$ and $\,(0,c)\,$.
  • Note: the foci are on the $\color{red}{y}$-axis, and the bigger number ($\,a^2 > b^2\,$) is beneath the $\,\color{red}{y}^2\,$.

Alternate Approach for Foci on the $y$-axis: Reflecting or Rotating

Once we have the equation for foci on the $x$-axis $\displaystyle\,\left(\frac{x^2}{a^2} +\frac{y^2}{b^2} = 1\right)\,$,
there are shorter and easier ways to get the equation for foci on the $y$-axis $\displaystyle\,\left(\frac{x^2}{b^2} +\frac{y^2}{a^2} = 1\right)\,$.
We didn't really have to go through this derivation again (although it's good practice).


Here are two ways.
For ease of reference, let $\,\cal G\,$ denote the graph of $\displaystyle\,\frac{x^2}{a^2} +\frac{y^2}{b^2} = 1\,$.
The two transformations used below are discussed in greater detail in this earlier optional lesson.

  1. Reflect $\,\cal G\,$ about the line $\,y = x\,$:
    Reflection about the line $\,y = x\,$ switches the coordinates of a point:   $\,(x,y)\,$ moves to $\,(y,x)\,$.

    In an equation, this is accomplished by switching the variables $\,x\,$ and $\,y\,$.

    Switching $\,x\,$ and $\,y\,$ turns $\displaystyle\,\frac{x^2}{a^2} +\frac{y^2}{b^2} = 1\,$ into $\displaystyle\,\frac{y^2}{a^2} +\frac{x^2}{b^2} = 1\,$.

    Done!
  2. Rotate $\,\cal G\,$ counterclockwise by $\,90^\circ\,$:
    Counterclockwise rotation by $\,90^\circ\,$ moves the point $\,(x,y)\,$ to the new point $\,(-y,x)\,$.

    In an equation, this is accomplished by:
    • replacing every $\,x\,$ by $\,y\,$
    • replacing every $\,y\,$ by $\,-x\,$

    These replacements turn $\displaystyle\,\frac{x^2}{a^2} +\frac{y^2}{b^2} = 1\,$ into: $$ \begin{gather} \cssId{s88}{\frac{y^2}{a^2} +\frac{(-x)^2}{b^2} = 1}\cr\cr \cssId{s89}{\frac{y^2}{a^2} + \frac{x^2}{b^2} = 1} \end{gather} $$ Easy!


Note:
The exercises in this lesson cover ellipses with centers at the origin
and both foci along the $x$-axis and foci along the $y$-axis.
Master the ideas from this section
by practicing the exercise at the bottom of this page.


When you're done practicing, move on to:
summary: equations of ellipses in standard form


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