Foci of an Ellipse Calculator

What Are the Foci of an Ellipse?

The foci (plural of focus) of an ellipse are two fixed points, $F_1$ and $F_2$, located along the ellipse's major axis such that the sum of the distances from any point $P(x, y)$ on the ellipse to both foci is always constant and equal to the major axis length $2a$:

$$d(P, F_1) + d(P, F_2) = 2a$$

This fundamental focal property defines Keplerian planetary orbits, whispering galleries in acoustics, and optical reflectors in physics and engineering.

How to Calculate the Focal Distance $c$

To locate the foci, you first find the linear eccentricity $c$, which represents the distance from the center $(h, k)$ to either focus along the major axis.

Using the Pythagorean relationship in an ellipse where $a \ge b$:

$$c^2 = a^2 - b^2 \implies c = \sqrt{a^2 - b^2}$$

$a$: semi-major axis length (half the longest diameter).
$b$: semi-minor axis length (half the shortest diameter).
$c$: focal distance from the center $(h, k)$.
$2c$: total separation distance between the two foci.

Foci Coordinates Formulas

1. Horizontal Ellipse (Major Axis Parallel to the x-Axis)

When the major axis is horizontal, the standard form equation is:

$$\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$$

The foci lie along the horizontal line $y = k$:

$$F_1 = (h - c, k), \quad F_2 = (h + c, k)$$

2. Vertical Ellipse (Major Axis Parallel to the y-Axis)

When the major axis is vertical, the standard form equation is:

$$\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1$$

The foci lie along the vertical line $x = h$:

$$F_1 = (h, k - c), \quad F_2 = (h, k + c)$$

Related Geometric Properties of the Ellipse

Eccentricity ($e$): Ratio of focal distance to semi-major axis, $e = \frac{c}{a} = \sqrt{1 - \frac{b^2}{a^2}}$. For an ellipse, $0 \le e < 1$. When $e = 0$, $a = b$ and the ellipse becomes a circle with both foci coinciding at the center.
Directrices: Two parallel lines situated at distance $d = \frac{a}{e} = \frac{a^2}{c}$ from the center. For a horizontal ellipse, $x = h \pm \frac{a^2}{c}$.
Latus Rectum: The chord passing through either focus perpendicular to the major axis, with length $\frac{2b^2}{a}$.
Center and Standard Form: You can convert general polynomial equations into canonical form using our Ellipse Standard Form Calculator or find center coordinates with the Center of Ellipse Calculator.
Circumference and Area: Compute overall dimensions and perimeter using our Ellipse Calculator and Ellipse Circumference Calculator.

Step-by-Step Calculation Example

Find the foci of the horizontal ellipse centered at $(2, -3)$ with semi-major axis $a = 5$ and semi-minor axis $b = 3$:

Compute focal distance: $c = \sqrt{a^2 - b^2} = \sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4$.
Because the orientation is horizontal, shift along the x-axis:
- $F_1 = (h - c, k) = (2 - 4, -3) = (-2, -3)$
- $F_2 = (h + c, k) = (2 + 4, -3) = (6, -3)$
Distance between foci: $2c = 2(4) = 8$.
Eccentricity: $e = c / a = 4 / 5 = 0.8$.

Frequently Asked Questions

How many foci does an ellipse have?

An ellipse has exactly 2 foci. When an ellipse degenerates into a perfect circle, the two foci converge and overlap at a single point (the center), where eccentricity $e = 0$.

What is the relationship between a, b, and c in an ellipse?

In any ellipse with semi-major axis $a$ and semi-minor axis $b$, the focal distance $c$ satisfies $a^2 = b^2 + c^2$, which means $c = \sqrt{a^2 - b^2}$.

How do I know if an ellipse is horizontal or vertical from its equation?

In standard form $\frac{(x-h)^2}{A} + \frac{(y-k)^2}{B} = 1$, compare the denominators $A$ and $B$. If $A > B$, the major axis is horizontal and $a^2 = A$. If $B > A$, the major axis is vertical and $a^2 = B$.

What happens to the foci when eccentricity increases toward 1?

As eccentricity $e \to 1$, the ellipse becomes increasingly elongated and flattened, and the foci move closer to the outer vertices. In the limit $e = 1$, the curve opens into a parabola with one finite focus.

Report