Equation of Each Ellipse and Finding the Foci, Vertices, and Co– Vertices of Ellipses
To write the equation of an ellipse, we need the parameters that will be explained in this article.

An Ellipse is a closed curve formed by a plane. There are two types of ellipses: Horizontal and Vertical
- If major axis of an ellipse is parallel to \(x\), its called horizontal ellipse.
- If major axis of an ellipse is parallel to \(y\), its called vertical ellipse.
Step by Step Guide to Find Equation of Ellipses
The standard form of the equation of an Ellipse is:
- Horizontal: \(\color{blue}{\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1}\)
- Vertical: \(\color{blue}{\frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1}\)
The center is: \(\color{blue}{(h, k)}\)
The vertices are: \(\color{blue}{(h+a, k), (h-a, k)}\)
The foci are: \(\color{blue}{(h+c, k), (h-c, k)}\), where \(\color{blue}{c=\sqrt{a^2-b^2}}\)
The Values can be calculated according to the standard form of the equation of ellipses.

Equation of Each Ellipse and Finding the Foci, Vertices, and Co– Vertices of Ellipses – Example 1:
Find the center, vertices, and foci of this ellipse: \(\frac{(x-2)^2}{36}+\frac{(y+4)^2}{16}=1\)
Solution:
The standard form of the equation of an Ellipse is: \(\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1\)
Then, \((h=2, k=-4, a=6, b=4)\).
So, the center is \((2, -4)\).
The vertices are \((h+a, k), (h-a, k) →(8, 4), (-4, 4)\)
Evaluate \(c\): \(c=\sqrt{a^2-b^2}\) \(=\sqrt{36-16}=2\sqrt{5}\)
Then the foci are \((2+2\sqrt{5}, -4)\) and \((2-2\sqrt{5}, -4)\).
Exercises for Equation of Finding the Foci, Vertices, and Co– Vertices of Ellipses
Find the center, vertices, and foci of each ellipse.
- \(\color{blue}{9x^2+4y^2=1}\)
- \(\color{blue}{16x^2+25y^2=100}\)
- \(\color{blue}{25x^2+4y^2+100x-40y=400}\)
- \(\color{blue}{\frac{(x-1)^2}{9}+\frac{y^2}{5}=100}\)

- \(\color{blue}{Center: (0, 0), Vertices: (0,\frac{1}{2}), (0, -\frac{1}{2}), foci: (0, \frac{\sqrt{5}}{6}), (0, -\frac{\sqrt{5}}{6})}\)
- \(\color{blue}{Center: (0, 0), Vertices: (\frac{5}{2}, 0), (-\frac{5}{2}, 0), foci: (\frac{3}{2}, 0), (-\frac{3}{2}, 0)}\)
- \(\color{blue}{Center: (-2, 5), Vertices: (-2,5+5\sqrt{6}), (-2, 5-5\sqrt{6}), foci: (-2, 5+3\sqrt{14}), (-2, 5-3\sqrt{14})}\)
- \(\color{blue}{Center: (1, 0), Vertices: (31, 0), (-29, 0), foci: (21, 0), (-19, 0)}\)
Related to This Article
More math articles
- The Ultimate PERT Math Formula Cheat Sheet
- How to Find the Volume of Cones and Pyramids? (+FREE Worksheet!)
- 4th Grade SBAC Math Worksheets: FREE & Printable
- Which Test Should You Take: GED, TASC, or HiSET?
- Word Problems Involving Volume of Cubes and Rectangular Prisms
- How to Simplify Radical Expressions? (+FREE Worksheet!)
- The Complete Guide to 5 Best GED Math Study Guides
- What Kind of Math Is on the CHSPE Test?
- 10 Most Common 4th Grade ACT Aspire Math Questions
- Top 10 SSAT Upper Level Math Practice Questions
What people say about "Equation of Each Ellipse and Finding the Foci, Vertices, and Co– Vertices of Ellipses"?
No one replied yet.