1. What is the Ellipse Foci Formula?

The ellipse foci formula calculates the distance from the center to each focus of an ellipse. For an ellipse with semi-major axis a and semi-minor axis b, the foci distance c is given by \( c = \sqrt{a^2 - b^2} \).

2. How Does the Calculator Work?

The calculator uses the ellipse foci formula:

Where:

• \( a \) — Semi-major axis (longer radius)
• \( b \) — Semi-minor axis (shorter radius)
• \( c \) — Distance from center to each focus

Explanation: The formula derives from the geometric definition of an ellipse, where the sum of distances from any point on the ellipse to the two foci is constant and equal to 2a.

3. Importance of Foci Calculation

Details: Calculating foci is essential in astronomy (planetary orbits), engineering (elliptical designs), and physics (wave propagation). The foci determine many properties of the ellipse, including its eccentricity.

4. Using the Calculator

Tips: Enter both semi-major (a) and semi-minor (b) axes in the same units. The semi-major axis must be larger than the semi-minor axis for a valid ellipse.

5. Frequently Asked Questions (FAQ)

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

Q2: Can b be larger 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 practical applications of ellipse foci?
A: Ellipse foci are used in satellite dish design, whispering galleries, planetary orbit calculations, and medical equipment like lithotripters.

Q4: How is eccentricity related to foci?
A: Eccentricity e = c/a, where 0 ≤ e < 1. A circle has e = 0, while a highly elongated ellipse approaches e = 1.

Q5: Can this formula be used for hyperbolas?
A: No, hyperbolas use a different formula: \( c = \sqrt{a^2 + b^2} \).