About Ellipses
What is an Ellipse?
An ellipse is a closed curve on a plane, defined as the locus of all points such that the sum of the distances from two fixed points (called foci) is constant.
Formula for an Ellipse
The standard form equation of an ellipse centered at the origin is:
\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]
Where:
- \(a\) is the length of the semi-major axis
- \(b\) is the length of the semi-minor axis
Key Properties
- Center: The point (0,0) in standard form
- Vertices: Points (±a, 0) on the major axis
- Co-vertices: Points (0, ±b) on the minor axis
- Foci: Points (±c, 0) where c² = a² - b²
- Eccentricity (e): The ratio c/a, where 0 ≤ e < 1
Calculation Steps
- Determine the semi-major axis \(a\) and semi-minor axis \(b\)
- Calculate the eccentricity: \(e = \sqrt{1 - \frac{b^2}{a^2}}\)
- Calculate the focal distance: \(c = ae\)
- Calculate the area: \(A = \pi ab\)
- Calculate the perimeter (approximation): \(P \approx 2\pi\sqrt{\frac{a^2 + b^2}{2}}\)
Example
Let's create an ellipse with \(a = 4\) and \(b = 3\):
Equation: \(\frac{x^2}{16} + \frac{y^2}{9} = 1\)