Find the Foci of an Ellipse Calculator

Ellipse Foci Formula:

\[ c = \sqrt{a^2 - b^2} \] \[ \text{Foci at } (h \pm c, k) \text{ if horizontal} \] \[ \text{Foci at } (h, k \pm c) \text{ if vertical} \]

Semi-major axis (a):

units

Semi-minor axis (b):

units

Center x-coordinate (h):

Center y-coordinate (k):

Orientation:

Distance to foci (c):

Foci positions:

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1. What is an Ellipse's Foci?

The foci (singular: focus) of an ellipse are two fixed points inside the ellipse such that the sum of the distances from any point on the ellipse to the two foci is constant. They are fundamental to the geometric definition of an ellipse.

2. How Does the Calculator Work?

The calculator uses the ellipse foci formula:

\[ c = \sqrt{a^2 - b^2} \]

Where:

\( a \) — Length of semi-major axis (longer radius)
\( b \) — Length of semi-minor axis (shorter radius)
\( c \) — Distance from center to each focus
\( (h,k) \) — Coordinates of the ellipse's center

Explanation: The foci are located along the major axis, equidistant from the center. For horizontal ellipses, they're at (h±c,k); for vertical ellipses, at (h,k±c).

3. Importance of Foci Calculation

Details: Foci are essential in orbital mechanics (planetary orbits), acoustics (whispering galleries), and optics (elliptical mirrors). They determine the ellipse's shape and eccentricity.

4. Using the Calculator

Tips: Enter semi-major axis (a) and semi-minor axis (b) lengths (a ≥ b), center coordinates (default 0,0), and orientation. All values must be positive with a ≥ b.

5. Frequently Asked Questions (FAQ)

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

Q2: How is eccentricity related to foci?
A: Eccentricity e = c/a, measuring how "stretched" the ellipse is (0 for circle, approaching 1 for very elongated).

Q3: Can the foci be outside the ellipse?
A: No, foci are always inside the ellipse, along the major axis between the center and vertices.

Q4: What's the sum of distances property?
A: For any point P on the ellipse, distance(P,F₁) + distance(P,F₂) = 2a (length of major axis).

Q5: How are ellipse foci used in real life?
A: Used in satellite dish design, planetary orbit calculations, and architectural acoustics for optimal sound reflection.