What is the difference between focus and foci?

Focus (singular) and foci (plural) refer to the fixed points used to define conic sections. An ellipse has two foci, a parabola has one focus, and a hyperbola has two foci. The defining property of an ellipse is that the sum of distances to the two foci is constant for every point on the ellipse.

How are the foci of an ellipse used in physics?

The foci are fundamental in astronomy: Kepler's First Law states that planets orbit the Sun in elliptical orbits with the Sun at one focus. This means a planet's distance from the Sun varies throughout its orbit. The foci are also used in acoustics (elliptical rooms concentrate sound at the second focus from the first) and optics (elliptical mirrors reflect light from one focus to the other).

Foci Of Ellipse

Q: What are the foci of an ellipse?

The foci (plural of focus) of an ellipse are two fixed points inside the ellipse such that the sum of distances from any point on the ellipse to the two foci is constant. The foci lie on the major axis, equidistant from the center. The distance from the center to each focus is c = sqrt(a^2 - b^2), where a is the semi-major axis and b is the semi-minor axis.

Q: What happens if a = b in the foci calculation?

If a = b (semi-major and semi-minor axes are equal), then c = sqrt(a^2 - a^2) = 0, meaning the foci coincide at the center. This is the special case of a circle, which is an ellipse with zero eccentricity. A circle has all points equidistant from the center, which acts as both foci together.

Calculate the foci of an ellipse given semi-major axis a and semi-minor axis b. Free online ellipse foci calculator finds c = sqrt(a^2-b^2), eccentricity, and orientation.

Find ellipse foci

Semi-Major Axis (a):

Semi-Minor Axis (b):

About This Calculator

This Foci of Ellipse Calculator computes the focal distance, eccentricity, and orientation of an ellipse from its semi-major axis (a) and semi-minor axis (b). The foci are two special points inside the ellipse that define its shape: the sum of distances from any point on the ellipse to both foci is constant and equal to 2a.

The key formula used is c = sqrt(a^2 - b^2), where c is the distance from the center to each focus. The eccentricity e = c/a measures how stretched the ellipse is, from e = 0 (circle) to e approaching 1 (very elongated). The calculator automatically detects whether the ellipse is horizontally oriented (a >= b, foci at +/-c on x-axis) or vertically oriented (b > a, foci at +/-c on y-axis).

Ellipses and their foci are central to astronomy (planetary orbits follow Kepler's laws with the Sun at a focus), physics (ellipse properties in wave mechanics), engineering (elliptical gears and arches), and optics (elliptical mirrors and lenses). Understanding the foci is essential for geometry students at the high school and undergraduate level.

Frequently Asked Questions

What are the foci of an ellipse?

The foci of an ellipse are two fixed points inside the ellipse such that the sum of distances from any point on the ellipse to the two foci is constant. The distance from the center to each focus is c = sqrt(a^2 - b^2).

How do you find the foci of an ellipse?

Use the formula c = sqrt(a^2 - b^2), where a is the semi-major axis and b is the semi-minor axis. The foci are located at (+/-c, 0) for a horizontal ellipse or at (0, +/-c) for a vertical ellipse.

What is the eccentricity of an ellipse?

Eccentricity (e) measures how elongated an ellipse is, calculated as e = c/a = sqrt(1 - b^2/a^2). It ranges from 0 (perfect circle) to 1 (highly elongated).

How are the foci used in astronomy?

Kepler's First Law states that planets orbit the Sun in elliptical orbits with the Sun at one focus. This means a planet's distance from the Sun varies throughout its orbit.

What happens if a = b?

If a = b, then c = 0, meaning the foci coincide at the center. This is the special case of a circle, which is an ellipse with zero eccentricity.