Graph Ellipses from Foci

Graphing Ellipses from Foci

Now that we've covered the characteristics of ellipses, like the centers of ellipses, axes, vertices, and co-vertices, and foci, we are ready the graph ellipses from foci. Remember that an ellipse is a set of points in a plane, the sum of whose distances from two fixed points (the foci) is constant. We know the standard form of an ellipse and how to complete the square to get there.

If we are given foci of an ellipse, we can graph the ellipse. Say we have the equation of an ellipse:

$$\frac{x^2}{a^2} + {y^2}{b^2} = 1$$

$a$ is the length of the semi-major axis and $b$ is the length of the semi-minor axis. The major axis is the longest diameter of the ellipse, and the minor axis is perpendicular to it, passing through the center.

The distance between the foci is $2c$. For an ellipse, we can find $c$ using the formula:

$$c^2 = a^2 - b^2$$

Once you calculate $c$, you can plot the foci. Locate the foci $F_1$ and $F_2$ on the major axis, $2c$ units away from the center in opposite directions. The length of the major axis is $2a$, and the length of the minor axis is $2b$.

Draw the ellipse using the major and minor axes as guides. The distance from any point on the ellipse to each focus should add up to a constant value.

Example

Suppose we have an ellipse with foci $F_1 (-3,0)$ and $F_2 (3,0)$, and the length of the major axis is $8$.

The $y$-values are the same in the two coordinates, so we can guess that the foci are on the $x$-axis. This is a horizontal ellipse.

The major axis is $8$, so $2a=8 \rightarrow a=4$.

The distance between the foci is $2c = 6 \rightarrow c=3$.

Then, we have the information we need to find $b$:

$$\begin{align} c^2 &= a^2 + b^2 \\ 3^2 &= 4^2 - b^2 \\ 9 &= 16 - b^2 \\ b^2 &= 16-9 \\ b &= \sqrt7 \\ \end{align}$$

Now that we've found $b$, we can plot the foci. The major axis is $8$, so it extends $4$ units to the left of $F_1$ and $4$ units to the right of $F_2$. The minor axis has a length of $2\sqrt7$, so it extends $\sqrt7$ units above and below the $x$-axis.

Sketch the ellipse using these guides.

The equation of this ellipse is: $\frac{x^2}{16} + \frac{y^2}{7} = 1$

Now that you've learned how to graph ellipses from foci, try graphing ellipses from an equation. For further practice, consider solving word problems using ellipses.

(Video) Example of Graphing Ellipses from Foci

by Anusha Rahman

In this video by Anusha Rahman, you will learn how to graph ellipses from foci.

Graph Ellipses from Foci

An ellipse is a stretched-out circle with two special points, called foci (the plural of focus). The shape of an ellipse is made by all points where the total distance to both foci is always the same.

Using the following equation as an example, $$\frac{(x-1)^{2}}{25}+\frac{(y-2)^{2}}{9}=1$$

To graph an ellipse from the foci, the first step is to plot the two foci on your graph. Then, find the midpoint between the foci, which is the center of the ellipse.

Next, you want to figure out how far apart the foci are. The distance from the center to each focus is $c$. You'll also need to know the length of the major axis, which is the longest distance across the ellipse. The distance is $2a$.

We can now use the following formula to find $b$ (the distance from the center to the ends of the minor axis).

$$c^2 = a^2 - b^2$$

Once you know $a$ and $b$, you can sketch the ellipse! Start at the center, then move $a$ units left and right for the major axis. Move $b$ units up and down for the minor axis (or the other way around if it's vertical).

Connect these four points with a smooth oval shape. Make sure that the ellipse is wider along the major axis and that the foci stay inside the shape.