How to Find Minor Radius of Ellipse?

The minor radius of an ellipse is found by identifying the smaller value between 'a' and 'b' in its standard equation form: (x−h)²/a² + (y−k)²/b² = 1.

An ellipse is a closed curve for which the sum of the distances from any point on the curve to two fixed points (called foci) is constant. It has two primary radii: the major radius and the minor radius. These define the ellipse's shape and dimensions.

Understanding the Ellipse's Standard Equation

To accurately find the minor radius, it's crucial to understand the standard form of an ellipse's equation. The general equation of an ellipse centered at (h, k) is given by:

(x−h)²/a² + (y−k)²/b² = 1

Here's what each component represents:

• (h, k): The coordinates of the center of the ellipse.
• a: The distance from the center to the ellipse along the horizontal axis (if a² is under (x-h)²) or half the length of the major or minor axis.
• b: The distance from the center to the ellipse along the vertical axis (if b² is under (y-k)²) or half the length of the major or minor axis.

Key Insight from Reference:
The provided reference explicitly states: "The equation of an ellipse written in the form (x−h)²/a² + (y−k)²/b² = 1. The center is (h,k) and the larger of a and b is the major radius and the smaller is the minor radius."

Identifying the Minor Radius

Based on the standard equation and the definition, the process for identifying the minor radius is straightforward:

1. Locate a² and b²: In the standard form (x−h)²/a² + (y−k)²/b² = 1, identify the denominators a² and b².
2. Calculate a and b: Take the square root of a² to find a and the square root of b² to find b.
3. Compare a and b: Determine which of these two values (a or b) is smaller.
4. The Smaller Value is the Minor Radius: The smaller of a and b is, by definition, the minor radius of the ellipse.

Practical Example

Let's consider an example to illustrate:

Question: Find the minor radius of the ellipse given by the equation: (x-2)²/9 + (y+1)²/25 = 1

Solution:

1. Identify a² and b²:

   • From (x-2)²/9, we have a² = 9.
   • From (y+1)²/25, we have b² = 25.

2. Calculate a and b:

   • a = √9 = 3
   • b = √25 = 5

3. Compare a and b:

   • a = 3
   • b = 5
   • Comparing these values, 3 is smaller than 5.

4. Determine Minor Radius:

   • Therefore, the minor radius of the ellipse is 3. In this case, b=5 would be the major radius.

Here's a summary table for clarity:

Understanding how to find the minor radius is fundamental in various fields, from astronomy (describing planetary orbits) to engineering (designing elliptical gears or structures).