Ellipse (E)

An ellipse is a geometric shape that can be thought of as a stretched circle. It is a type of conic section that is formed by the intersection of a cone with a plane in a way that produces a closed curve. Ellipses are important in physics, astronomy, engineering, and many other fields.

Definition

An ellipse is the set of all points in a plane such that the sum of the distances from two fixed points (called the foci) is constant. This constant sum is greater than the distance between the foci, ensuring that the shape is elongated.

Standard Equation of an Ellipse

The standard equation of an ellipse with its center at the origin (0,0) and its major axis along the x-axis is given by:

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

where:

$a$ is the semi-major axis (half the length of the longest diameter)
$b$ is the semi-minor axis (half the length of the shortest diameter)

If the major axis is along the y-axis, the equation is:

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

Important Points and Differences

Property	Description
Center	The point equidistant from all points on the ellipse. For the standard ellipse, it is at the origin (0,0).
Foci	Two fixed points located on the major axis at a distance of $c$ from the center, where $c = \sqrt{a^2 - b^2}$.
Major Axis	The longest diameter of the ellipse, which passes through the center and both foci. Its length is $2a$.
Minor Axis	The shortest diameter of the ellipse, which passes through the center and is perpendicular to the major axis. Its length is $2b$.
Eccentricity	A measure of how much the ellipse deviates from being a circle. It is given by $e = \frac{c}{a}$, where $0 < e < 1$ for an ellipse.
Directrices	Lines perpendicular to the major axis that help define the ellipse. They are located at a distance of $\frac{a^2}{c}$ from the center.

Formulas Related to Ellipse

Area of an ellipse: $A = \pi a b$
Perimeter of an ellipse (approximate): $P \approx \pi \left[ 3(a + b) - \sqrt{(3a + b)(a + 3b)} \right]$
Length of the latus rectum (the chord parallel to the minor axis through a focus): $2b^2/a$

Examples

Example 1: Finding the Foci

Given an ellipse with a semi-major axis of 5 units and a semi-minor axis of 3 units, find the foci.

Solution:

Calculate $c$ using $c = \sqrt{a^2 - b^2}$.
$c = \sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4$.
The foci are at $(\pm c, 0)$, so they are at $(\pm 4, 0)$.

Example 2: Eccentricity of an Ellipse

Find the eccentricity of an ellipse with a semi-major axis of 6 units and a semi-minor axis of 4 units.

Solution:

Calculate $c$ using $c = \sqrt{a^2 - b^2}$.
$c = \sqrt{6^2 - 4^2} = \sqrt{36 - 16} = \sqrt{20} = 2\sqrt{5}$.
Calculate the eccentricity $e = \frac{c}{a}$.
$e = \frac{2\sqrt{5}}{6} = \frac{\sqrt{5}}{3}$.

Example 3: Area of an Ellipse

Calculate the area of an ellipse with a semi-major axis of 7 units and a semi-minor axis of 2 units.

Solution:

Use the area formula $A = \pi a b$.
$A = \pi \cdot 7 \cdot 2 = 14\pi$ square units.

In conclusion, an ellipse is a fascinating geometric shape with many interesting properties and formulas. Understanding the concepts of foci, major and minor axes, and eccentricity is crucial for solving problems related to ellipses. The standard equation of an ellipse is the starting point for analyzing its geometry, and the examples provided illustrate how to apply the formulas in practical situations.