Find Foci Calculator

Calculator

Enter the parameters of your ellipse or hyperbola to find its foci.

Visual representation of the conic section and its foci. (Scale adjusts)

Parameter	Value
Type	–
Center (h, k)	–
Semi-axis a	–
Semi-axis b	–
Orientation	–
c	–
Eccentricity (e)	–
Focus 1 (F1)	–
Focus 2 (F2)	–
Equation	–

Summary of input parameters and calculated results.

What is a Find Foci Calculator?

A Find Foci Calculator is a tool used to determine the coordinates of the foci (plural of focus) of a conic section, specifically an ellipse or a hyperbola, given its defining parameters. The foci are special points within these curves that are fundamental to their geometric definition and properties. For an ellipse, the sum of the distances from any point on the ellipse to the two foci is constant. For a hyperbola, the absolute difference of the distances from any point on the hyperbola to the two foci is constant.

This calculator is useful for students studying conic sections in mathematics (algebra, geometry, pre-calculus, calculus), engineers, physicists, and astronomers who work with elliptical or hyperbolic shapes and orbits. For instance, planets orbit the sun in elliptical paths with the sun at one focus.

Common misconceptions include thinking that parabolas have two foci (they have one focus and a directrix) or that the 'c' value is the same for ellipses and hyperbolas with the same 'a' and 'b' (the formulas differ).

Find Foci Calculator Formula and Mathematical Explanation

The location of the foci depends on the type of conic section (ellipse or hyperbola), its center (h, k), the lengths of its semi-major/transverse axis (a) and semi-minor/conjugate axis (b), and its orientation.

The distance from the center to each focus is denoted by 'c'.

For an Ellipse:

The standard equation depends on the orientation:

• Horizontal Major Axis: (x-h)²/a² + (y-k)²/b² = 1 (where a > b)
• Vertical Major Axis: (x-h)²/b² + (y-k)²/a² = 1 (where a > b)

In both cases, 'a' is the semi-major axis and 'b' is the semi-minor axis. The relationship between a, b, and c is:

c² = a² – b² => c = √(a² – b²) (since a > b, a² – b² > 0)

• If Horizontal Major Axis: Foci are at (h ± c, k)
• If Vertical Major Axis: Foci are at (h, k ± c)

The eccentricity 'e' of an ellipse is e = c/a (0 ≤ e < 1).

For a Hyperbola:

The standard equation depends on the orientation:

• Horizontal Transverse Axis: (x-h)²/a² – (y-k)²/b² = 1
• Vertical Transverse Axis: (y-k)²/a² – (x-h)²/b² = 1

Here, 'a' is the semi-transverse axis and 'b' is the semi-conjugate axis. The relationship between a, b, and c is:

c² = a² + b² => c = √(a² + b²)

• If Horizontal Transverse Axis: Foci are at (h ± c, k)
• If Vertical Transverse Axis: Foci are at (h, k ± c)

The eccentricity 'e' of a hyperbola is e = c/a (e > 1).

Variables Table:

Variable	Meaning	Unit	Typical Range
(h, k)	Coordinates of the center	Length units	Any real numbers
a	Semi-major axis (ellipse) or semi-transverse axis (hyperbola)	Length units	a > 0
b	Semi-minor axis (ellipse) or semi-conjugate axis (hyperbola)	Length units	b > 0 (for ellipse, a ≥ b)
c	Distance from center to focus	Length units	c ≥ 0
e	Eccentricity	Dimensionless	0 ≤ e < 1 (ellipse), e > 1 (hyperbola)

Practical Examples (Real-World Use Cases)

Let's see how our Find Foci Calculator works with some examples.

Example 1: Elliptical Orbit

A satellite orbits the Earth in an elliptical path with the Earth at one focus. The center of the ellipse is at (0, 0), the semi-major axis (a) is 10,000 km, and the semi-minor axis (b) is 9,900 km. The major axis is horizontal.

• Type: Ellipse
• Center (h, k): (0, 0)
• Semi-major axis (a): 10000
• Semi-minor axis (b): 9900
• Orientation: Horizontal

Using the formula c = √(a² – b²) = √(10000² – 9900²) = √(100,000,000 – 98,010,000) = √(1,990,000) ≈ 1410.67 km.

The foci are at (0 ± 1410.67, 0), so F1 = (1410.67, 0) and F2 = (-1410.67, 0). The Earth is at one of these foci.

Example 2: Hyperbolic Path

A comet follows a hyperbolic path around the Sun (which is at one focus). The center of the hyperbola is at (0, 0), the semi-transverse axis (a) is 2 AU (Astronomical Units), and the semi-conjugate axis (b) is 3 AU. The transverse axis is horizontal.

• Type: Hyperbola
• Center (h, k): (0, 0)
• Semi-transverse axis (a): 2
• Semi-conjugate axis (b): 3
• Orientation: Horizontal

Using the formula c = √(a² + b²) = √(2² + 3²) = √(4 + 9) = √(13) ≈ 3.606 AU.

The foci are at (0 ± 3.606, 0), so F1 = (3.606, 0) and F2 = (-3.606, 0). The Sun is at one of these foci.

How to Use This Find Foci Calculator

1. Select Conic Type: Choose either "Ellipse" or "Hyperbola".
2. Enter Center Coordinates: Input the values for 'h' (x-coordinate) and 'k' (y-coordinate) of the center of your conic section.
3. Select Orientation: Choose "Horizontal" or "Vertical" for the major axis (ellipse) or transverse axis (hyperbola).
4. Enter Semi-axes Lengths:
   • Input 'a' (semi-major/transverse axis). Ensure a > 0. For an ellipse, 'a' is associated with the selected orientation's axis and should be greater than or equal to 'b'.
   • Input 'b' (semi-minor/conjugate axis). Ensure b > 0.
5. Read the Results: The calculator will instantly display the coordinates of the two foci (F1 and F2), the distance 'c', the eccentricity 'e', and the standard equation of the conic section.
6. View Chart and Table: A visual representation and a summary table are provided for better understanding.

Use the "Reset" button to clear inputs to default values and "Copy Results" to copy the main findings.

Key Factors That Affect Foci Location

1. Center (h, k): The foci are located relative to the center. If the center moves, the foci move with it by the same amount.
2. Semi-major/transverse Axis (a): This determines the overall size along the main axis. A larger 'a' generally leads to foci further from the center (especially for hyperbolas or very eccentric ellipses).
3. Semi-minor/conjugate Axis (b): This influences the 'roundness' (ellipse) or 'openness' (hyperbola). For an ellipse, as 'b' approaches 'a', 'c' gets smaller, and the foci get closer to the center. For a hyperbola, a larger 'b' also increases 'c'.
4. Relationship between a and b: The difference (ellipse: a²-b²) or sum (hyperbola: a²+b²) directly determines c², and thus the distance 'c' to the foci.
5. Conic Type (Ellipse/Hyperbola): The formula for 'c' is different (subtraction for ellipse, addition for hyperbola), significantly impacting the foci location for the same 'a' and 'b'.
6. Orientation: This determines whether 'c' is added/subtracted from 'h' (horizontal) or 'k' (vertical) to find the foci coordinates.

Frequently Asked Questions (FAQ)

What is a focus in a conic section?

A focus is a special point used to define a conic section. Ellipses and hyperbolas have two foci.

How many foci does an ellipse have?

An ellipse has two foci.

How many foci does a hyperbola have?

A hyperbola has two foci.

Does a parabola have two foci?

No, a parabola has one focus and a directrix.

What happens if a = b in an ellipse?

If a = b in an ellipse, then c = √(a² – a²) = 0. The two foci coincide at the center, and the ellipse becomes a circle.

What does eccentricity 'e' tell us?

Eccentricity measures how much a conic section deviates from being circular. For an ellipse, 0 ≤ e < 1 (e=0 is a circle). For a parabola, e=1 (though not handled by this calculator). For a hyperbola, e > 1.

Can 'a' be smaller than 'b' when I input for an ellipse?

If you select "Horizontal" orientation, 'a' is assumed along the x-direction and 'b' along y. If a < b, the major axis is actually vertical. Our calculator uses 'a' as the semi-axis along the chosen orientation, so if horizontal is chosen with a < b, it will calculate as if 'a' were the larger one in the formula c=sqrt(a^2-b^2) if you meant a was semi-major. It's best to input 'a' as the larger semi-axis for ellipse and select the corresponding orientation, or ensure a>=b for the given orientation if 'a' refers to the semi-axis along that direction.

Where are the foci used in real life?

Elliptical reflectors use the property that light/sound from one focus reflects to the other (e.g., whispering galleries, medical lithotripsy). Planetary orbits are ellipses with the star at one focus. Hyperbolic paths are taken by some comets or in navigation systems (LORAN).