What is the formula for the major axis of an ellipse?

The formula for the length of the major axis of an ellipse is 2a, where 'a' represents the length of the semi-major axis. This 'a' value is crucial in defining the ellipse's shape and orientation within its standard equation.

Understanding the Major Axis in Ellipse Formulas

The major axis of an ellipse is its longest diameter, passing through the center of the ellipse and both foci. It is a fundamental component in defining the ellipse's overall dimensions and orientation.

• Semi-major Axis (a): This is half the length of the major axis, extending from the center of the ellipse to a vertex along the major axis. In the standard equation of an ellipse, 'a' is always the larger of the two semi-axis lengths (compared to 'b', the semi-minor axis).

Standard Forms and Major Axis Orientation

The standard form of an ellipse centered at (h, k) is key to understanding the major axis. The position of a² (or b²) determines whether the major axis is horizontal or vertical.

The provided reference states:

• "If the y-coordinates of the given vertices and foci are the same, then the major axis is parallel to the x-axis. Use the standard form (x−h)2/a2 + (y−k)2/b2 = 1."
• "If the x-coordinates of the given vertices and foci are the same, then the major axis is parallel to the y-axis." (In this case, a² would be under the (y-k)² term, making it a vertical ellipse).

This means that 'a' is always associated with the major axis, regardless of its orientation. The larger denominator in the standard equation determines a², and its position (under (x-h)² or (y-k)²) determines the orientation.

Here's a breakdown of the standard forms and their corresponding major axis properties:

Determining Major Axis Orientation and Equation (Practical Insights)

Understanding the major axis involves identifying its length and its alignment (horizontal or vertical) relative to the coordinate axes.

• Case 1: Horizontal Major Axis

  • Condition: As per the reference, if the y-coordinates of the vertices and foci are the same, the major axis is horizontal (parallel to the x-axis).
  • Standard Form: (x−h)²/a² + (y−k)²/b² = 1 (where a² is the larger denominator and is under the (x-h)² term).
  • Equation of Major Axis Line: The line containing the major axis is y = k, where (h, k) is the center of the ellipse.

• Case 2: Vertical Major Axis

  • Condition: As per the reference, if the x-coordinates of the vertices and foci are the same, the major axis is vertical (parallel to the y-axis).
  • Standard Form: (x−h)²/b² + (y−k)²/a² = 1 (where a² is the larger denominator and is under the (y-k)² term).
  • Equation of Major Axis Line: The line containing the major axis is x = h, where (h, k) is the center of the ellipse.

Example: Finding the Major Axis Length and Orientation

Let's consider an ellipse with the equation:
(x - 3)² / 49 + (y + 2)² / 25 = 1

1. Identify a² and b²:
   • Compare the denominators: 49 and 25.
   • Since 49 is the larger value, a² = 49, and b² = 25.
2. Calculate 'a':
   • a = √49 = 7.
3. Calculate Major Axis Length:
   • The length of the major axis is 2a = 2 * 7 = 14.
4. Determine Orientation:
   • Since a² (which is 49) is under the (x - h)² term, the major axis is horizontal (parallel to the x-axis).
5. Identify Center (h, k):
   • From (x - 3)² and (y + 2)², the center (h, k) is (3, -2).
6. Equation of Major Axis Line:
   • For a horizontal ellipse, the major axis lies along the line y = k, so y = -2.

In summary, the fundamental formula for the length of an ellipse's major axis is 2a, derived directly from its standard form where 'a' always represents the semi-major axis length, with its position indicating the axis's orientation.