1. What is an Ellipse's Focal Point?

An ellipse has two focal points (foci) that are symmetrically located along its major axis. The sum of the distances from any point on the ellipse to both foci is constant and equal to the length of the major axis.

2. How Does the Calculator Work?

The calculator uses the ellipse focal point formula:

Where:

• \( c \) — Distance from center to each focus (length units)
• \( a \) — Semi-major axis length (length units)
• \( b \) — Semi-minor axis length (length units)

Explanation: The focal distance is calculated from the relationship between the semi-major and semi-minor axes of the ellipse.

3. Importance of Focal Points

Details: The focal points are fundamental to an ellipse's definition and properties. They determine the ellipse's shape and are crucial in applications like planetary orbits (Kepler's laws) and optical systems.

4. Using the Calculator

Tips: Enter the semi-major axis (a) and semi-minor axis (b) in consistent length units. The semi-major axis must be longer than the semi-minor axis (a > b).

5. Frequently Asked Questions (FAQ)

Q1: What if a equals b?
A: When a = b, the ellipse becomes a circle and the foci coincide at the center (c = 0).

Q2: Can b be greater than a?
A: No, by definition a is always the semi-major (longer) axis and b is the semi-minor (shorter) axis.

Q3: What are some real-world applications?
A: Elliptical orbits in astronomy, whispering galleries, elliptical gears, and satellite dish design.

Q4: How are focal points related to eccentricity?
A: Eccentricity e = c/a, where 0 ≤ e < 1 for an ellipse (e=0 for a circle).

Q5: Can this calculator handle different units?
A: Yes, as long as both a and b are in the same units, the result will be in those units.