Graphing Ellipses

HOW TO

Given the standard form of an equation for an ellipse centered at  $(0,0)$,  sketch the graph.

1. Use the standard forms of the equations of an ellipse to determine the major axis, vertices, co-vertices, and foci.
   1. If the equation is in the form $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$,  where $a > b$, then
      • the major axis is the $x$-axis
      • the coordinates of the vertices are $(±a,0)$
      • the coordinates of the co-vertices are $(0,±b)$
      • the coordinates of the foci are $(±c,0)$
   2. If the equation is in the form $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$, where $a > b$, then
      • the major axis is the $y$-axis
      • the coordinates of the vertices are $(0,±a)$
      • the coordinates of the co-vertices are $(±b,0)$
      • the coordinates of the foci are $(0,±c)$
2. Solve for $c$ using the equation $c^2=a^2−b^2$.
3. Plot the center, vertices, co-vertices, and foci in the coordinate plane, and draw a smooth curve to form the ellipse.

Example 3

Graphing an Ellipse Centered at the Origin

Graph the ellipse given by the equation, $\frac{x^2}{9}+\frac{y^2}{25}=1$. Identify and label the center, vertices, co-vertices, and foci.

Solution

First, we determine the position of the major axis. Because $25 > 9$, the major axis is on the $y$-axis. Therefore, the equation is in the form $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$, where $b^2=9$ and $a^2=25$. It follows that:

• the center of the ellipse is $(0,0)$
• the coordinates of the vertices are $(0,±a)=(0,±\sqrt{25})=(0,±5)$
• the coordinates of the co-vertices are $(±b,0)=(±\sqrt{9},0)=(±3,0)$
• the coordinates of the foci are $(0,±c)$, where $c^2=a^2−b^2$ Solving for $c$, we have:

$\begin{aligned}c &= ± \sqrt{a^2 - b^2} \\&= ± \sqrt{25-9} \\&= ± \sqrt{16} \\&= ±4\end{aligned}$

Therefore, the coordinates of the foci are $(0,±4)$.

Next, we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse. See Figure 8.

Figure 8

Try It #3

Graph the ellipse given by the equation $\frac{x^2}{36}+\frac{y^2}{4}=1$. Identify and label the center, vertices, co-vertices, and foci.

Example 4

Graphing an Ellipse Centered at the Origin from an Equation Not in Standard Form

Graph the ellipse given by the equation $4x^2+25y^2=100$. Rewrite the equation in standard form. Then identify and label the center, vertices, co-vertices, and foci.

Solution

First, use algebra to rewrite the equation in standard form.

$\begin{aligned}4x^2 &+25y^2 = 100 \\\frac{4x^2}{100} &+25y^2 = \frac{100}{100} \\\frac{x^2}{25} &+ \frac{y^2}{4} = 1\end{aligned}$

Next, we determine the position of the major axis. Because $25 > 4$, the major axis is on the $x$-axis. Therefore, the equation is in the form $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, where $a^2=25$ and $b^2=4$. It follows that:

• the center of the ellipse is $(0,0)$
• the coordinates of the vertices are $(±a,0)=(±\sqrt{25},0)=(±5,0)$
• the coordinates of the co-vertices are $(0,±b)=(0, ±\sqrt{4})=(0,±2)$
• the coordinates of the foci are $(±c,0)$, where $c^2=a^2−b^2$ Solving for $c$, we have:

$\begin{aligned}c &= ± \sqrt{a^2 - b^2} \\&= ± \sqrt{25 - 4} \\&= ± \sqrt{21}\end{aligned}$

Therefore, the coordinates of the foci are $(±\sqrt{21}, 0)$.

Next, we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse.

Figure 9

Try It #4

Graph the ellipse given by the equation $49x^2+16y^2=784$.  Rewrite the equation in standard form. Then identify and label the center, vertices, co-vertices, and foci.