Ellipses (standard form, foci, center at (h, k))
Ellipses
Ellipses is a topic in Conic Sections in the California Common Core State Standards. It is aligned to Standard G-GPE.3, which requires students to derive the equation of an ellipse from its foci.
An ellipse is the set of points whose distances to two foci add to a constant, with standard form \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\).
Theory
An ellipse is the set of points whose distances to two fixed foci add to a constant. Centered at the origin its standard form is
The major axis (length \(2a\)) lies along the variable with the larger denominator; the minor axis has length \(2b\). The foci sit a distance \(c\) from the center, where
Standard form and the focal distance:
How to analyze an ellipse
- Identify \(a^2\) and \(b^2\) (\(a^2\) is the larger).
- Major axis lies along the larger-denominator variable, length \(2a\).
- Foci: \(c=\sqrt{a^2-b^2}\) from the center, along the major axis.
- Center from the \((x-h),(y-k)\) shifts.
\(a^2=25,\ b^2=9\), so \(a=5,\ b=3\); \(c^2=25-9=16\).
| \(\text{major axis}\) | \(=\) | \(2a=10\) |
| \(\text{minor axis}\) | \(=\) | \(2b=6\) |
| \(\text{foci}\) | \(=\) | \((\pm 4,0)\) |
Here the larger denominator is under \(y^2\), so the major axis is vertical.
| \(a=5\ (\text{on }y),\ b=3\) | ||
| \(\text{foci}\) | \(=\) | \((0,\pm 4)\) |
Place \(a^2\) under \(x^2\).
Read the shifts directly.
| \(\text{center}\) | \(=\) | \((2,-1)\) |
Common pitfalls
Frequently asked questions
What is an ellipse?
The set of points whose distances to two foci sum to a constant; its standard form is \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\).
How do you find the foci of an ellipse?
Compute \(c=\sqrt{a^2-b^2}\); the foci lie a distance \(c\) from the center along the major axis.
How do you tell which axis is major?
The larger denominator marks the major axis. If it is under \(y^2\), the ellipse is taller; under \(x^2\), it is wider.
What is the equation of a shifted ellipse?
\(\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1\), centered at \((h,k)\).