Standard form

Understanding the Standard Form of an Ellipse

An ellipse is a set of all points in a plane where the sum of the distances from two fixed points (foci) is constant. The standard form of an ellipse's equation is a way to express this geometric figure algebraically and analyze its properties.

The Standard Form Equations

There are two standard form equations for an ellipse, depending on its orientation:

  1. Horizontal Major Axis:

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

  1. Vertical Major Axis:

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

In these equations, $(h, k)$ is the center of the ellipse, $a$ is the semi-major axis length, and $b$ is the semi-minor axis length. For a horizontal ellipse, $a > b$, and for a vertical ellipse, $a < b$.

Key Components

Component Description
Center $(h, k)$ The point at the middle of the ellipse.
Semi-Major Axis ($a$) Half the length of the longest diameter of the ellipse.
Semi-Minor Axis ($b$) Half the length of the shortest diameter of the ellipse.
Foci ($F_1$ and $F_2$) Two fixed points inside the ellipse used to define it.
Eccentricity ($e$) A measure of how much the ellipse deviates from being circular, calculated as $e = \sqrt{1 - \frac{b^2}{a^2}}$ for a horizontal ellipse and $e = \sqrt{1 - \frac{a^2}{b^2}}$ for a vertical ellipse.

Formulas Related to Ellipses

  • Area of an Ellipse:

[ A = \pi a b ]

  • Circumference of an Ellipse (Approximation):

[ C \approx \pi \left[ 3(a + b) - \sqrt{(3a + b)(a + 3b)} \right] ]

  • Distance between Foci:

[ c = 2e = 2\sqrt{a^2 - b^2} \text{ for a horizontal ellipse} ] [ c = 2e = 2\sqrt{b^2 - a^2} \text{ for a vertical ellipse} ]

Examples

Example 1: Identifying the Standard Form

Given the equation:

[ 9(x-2)^2 + 4(y+3)^2 = 36 ]

Identify the standard form and the key components of the ellipse.

Solution:

First, divide the equation by 36 to get the standard form:

[ \frac{(x-2)^2}{4} + \frac{(y+3)^2}{9} = 1 ]

This is a vertical ellipse since the larger denominator is under the $y$-term. The key components are:

  • Center: $(h, k) = (2, -3)$
  • Semi-Major Axis: $a = 3$
  • Semi-Minor Axis: $b = 2$
  • Eccentricity: $e = \sqrt{1 - \frac{2^2}{3^2}} = \sqrt{1 - \frac{4}{9}} = \sqrt{\frac{5}{9}} = \frac{\sqrt{5}}{3}$
  • Foci: $c = 2\sqrt{3^2 - 2^2} = 2\sqrt{5}$, so the foci are at $(2, -3 \pm \sqrt{5})$

Example 2: Finding the Area

Given an ellipse with the standard form equation:

[ \frac{(x-1)^2}{16} + \frac{(y+2)^2}{25} = 1 ]

Find the area of the ellipse.

Solution:

Identify $a$ and $b$:

  • $a = 5$
  • $b = 4$

Use the area formula:

[ A = \pi a b = \pi \cdot 5 \cdot 4 = 20\pi ]

The area of the ellipse is $20\pi$ square units.

Understanding the standard form of an ellipse is crucial for solving problems related to its geometry and for performing calculations involving its area, circumference, and other properties. By mastering the standard form, you can easily identify the key components of an ellipse and apply the appropriate formulas.