Ellipses

Adapted from OpenStax College Algebra 2e¹

Equation of an ellipse

The equation of an ellipse, in standard form, is

$\frac{(x - h)^{2}}{a^{2}} + \frac{(y - k)^{2}}{b^{2}} = 1 .$

Drawing an ellipse

Center

Just like in drawing a circle, we draw an ellipse by first identifying its center. For an equation of standard form above, the center has coordinates $(h, k) .$

Vertices

We then locate the vertices, the four points furthest from the center horizontally and vertically. The two vertices on the left and right are $a$ units from the center, so their coordinates are $(h - a, k)$ and $(h + a, k) .$

Meanwhile, the two vertices on the top and bottom are $b$ units from the center, so their coordinates are $(h, k - b)$ and $(h, k + b) .$

Marking out these points then allow us to draw the ellipse.

Example

The following is a sketch of the ellipse with equation

$\frac{(x + 2)^{2}}{4} + \frac{(y - 5)^{2}}{9} = 1$

ellipses

Content in this page is adapted from OpenStax College Algebra 2e by Jay Abramson under the Creative Commons Attribution License.
Access for free at https://openstax.org/books/college-algebra-2e/pages/5-6-rational-functions ↩︎