Detailed Notes on Graphing Ellipses

Introduction to Graphing an Ellipse

  • Ellipse Definition: An ellipse is a set of points in a plane, the sum of whose distances from two fixed points (foci) is constant.

1. Graphing an Ellipse Centered at the Origin Given its Standard Form

Standard Form Equations

  • If the ellipse is in the form x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, where a > b:

    • Major Axis: Along the x-axis
    • Vertices: Located at (±a, 0)
    • Co-Vertices: Located at (0, ±b)
    • Foci: Located at (±c, 0) where c=a2−b2c = \sqrt{a^2 - b^2}

  • If the ellipse is in the form x2b2+y2a2=1\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1, where a > b:

    • Major Axis: Along the y-axis
    • Vertices: Located at (0, ±a)
    • Co-Vertices: Located at (±b, 0)
    • Foci: Located at (0, ±c) where c=a2−b2c = \sqrt{a^2 - b^2}
Steps for Graphing
  1. Identify Parameters: From the standard form, determine a, b, and c.
  2. Calculate c: Use the formula c2=a2−b2c^2 = a^2 - b^2 to find the foci.
  3. Plot Points: Mark the center (0,0), vertices, co-vertices, and foci on the coordinate plane.
  4. Draw Ellipse: Sketch a smooth curve connecting the vertices and co-vertices.

2. Illustrative Examples

Example 1: Graph the Ellipse x29+y225=1\frac{x^2}{9} + \frac{y^2}{25} = 1
  • Identifying parameters:
    • Major axis: y-axis (since 25 > 9)
    • a2=25a^2 = 25, b2=9b^2 = 9; hence a=5a = 5, b=3b = 3
  • Coordinates:
    • Center: (0, 0)
    • Vertices: (0, ±5)
    • Co-Vertices: (±3, 0)
    • Foci: (0, ±4)
Example 2: Graph the Ellipse x236+y24=1\frac{x^2}{36} + \frac{y^2}{4} = 1
  • Parameters:
    • Major axis: x-axis (since 36 > 4)
    • a2=36a^2 = 36, b2=4b^2 = 4; hence a=6a = 6, b=2b = 2
  • Coordinates:
    • Center: (0, 0)
    • Vertices: (±6, 0)
    • Co-Vertices: (0, ±2)
    • Foci: (±√32, 0)

3. Graphing an Ellipse Centered at the Origin Given its General Form

Steps for Conversion
  1. Rewrite to Standard Form: Ensure the equation resembles the standard forms of an ellipse.
  2. Identify Major Axis and Parameters: Analyze the rewritten equation to find a and b.
  3. Follow Previous Steps: Plot the identified points and draw.
Example 3: Given the General Form 4x2+25y2=1004x^2 + 25y^2 = 100
  • Rewrite: Conversion yields x225+y24=1\frac{x^2}{25} + \frac{y^2}{4} = 1
  • Parameters:
    • Major axis: y-axis (25 > 4)
    • a2=25a^2 = 25, b2=4b^2 = 4; hence a=5a = 5, b=2b = 2
    • Coordinates:
    • Center: (0, 0)
    • Vertices: (0, ±5)
    • Co-Vertices: (±2, 0)
    • Foci: (0, ±√21)
General Notes
  • Symmetry: Elipses are symmetrical about both axes.
  • Applications: Useful in astronomy, engineering, and other fields involving shapes,
    and paths.

Conclusion

  • Understanding the process of graphing ellipses is crucial for further studies in geometry and higher mathematics. Correct identification of parameters is key to accurately representing an ellipse on the coordinate plane.