5.6.3

Introduction to Ellipses

• Exploration of the equation and graph of the conic section called an ellipse created by a plane intersecting a cone at an angle.

Generation of the Ellipse

• An ellipse can be generated when a plane intersects a double napped cone at various angles.
• The position of the plane can be adjusted vertically, which affects the size of the ellipse.

Mathematical Representation of the Ellipse

• The standard equation of an ellipse in relation to the axes is:   $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$
    - Where the parameters are defined as follows:     - Origin: Located at (0, 0) in the standard form.     - a: Distance from the origin to the edge of the ellipse along the x-axis (horizontal distance).     - b: Distance from the origin to the edge of the ellipse along the y-axis (vertical distance).

• A more versatile form of the equation is given by:   $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$
    - Where:     - (h, k): Represents the center (or origin) of the ellipse in the Cartesian plane.

Graphing an Ellipse

• The following steps outline how to graph an ellipse based on its equation:   1. Identify the equation: Compare it with the standard form to establish the parameters.   2. Determine the center: Given the standard form, extract values of h and k to find the center: (h, k).   3. Value of a and b: Identify the values of a and b directly from the equation:      - Example: For $\frac{(x-2)^2}{3^2} + \frac{(y-3)^2}{2^2} = 1$:        - h = 2, k = 3, a = 3, b = 2

Steps to Graph a Given Ellipse

• Example equation: Determine the origin and values of a and b:   - Standard form comparison reveals:     - Origin: (4, -2)     - Values: a = 5, b = 3

1. Plot the center: Mark the point at the origin (4, -2).
2. Plot the horizontal points: Count horizontally to the left and right from the center (4, -2):     - Points at (9, -2) and (-1, -2).
3. Plot the vertical points: Count vertically from the center:     - Points at (4, 1) and (4, -5).
4. Connect the points: Draw the ellipse through these points.

Additional Example for Practice

• Given another example: Identify and graph the ellipse from:   - Equation: Identify the origin and draw the ellipse based on:     - Origin: (-1, 2)     - For a = 1, plot edge points at (-2, 2) and (0, 2).     - For b = 5, plot upper and lower points at (-1, 7) and (-1, -3).
• Connect all six points to form the ellipse.

Writing the Equation of an Ellipse

1. Identify origin: Identify the origin of the ellipse as (2, 1).
2. Determine a and b:   - For a=4, validate by counting right and left from the center.   - For b=2, validate by counting up and down from the center.
3. Formulate the equation:    - Start with the standard form and fill h, k, a, and b:    - Equation becomes: $\frac{(x-2)^2}{4^2} + \frac{(y-1)^2}{2^2} = 1$

Summary

• An ellipse can be visualized and mathematically represented through specific equations determined by the orientation of a plane to a conical surface. The parameters and their geometric implications provide a comprehensive understanding for graphing and deriving equations of ellipses.