Finding Ellipse Equation Calculator

Enter the center coordinates (h, k) and the radii (rx, ry) to find the standard equation of the ellipse.

Ellipse Properties:

Property	Value
Center (h, k)	(0, 0)
Horizontal Radius (rx)	5
Vertical Radius (ry)	3
Semi-major axis (a)	5
Semi-minor axis (b)	3
Focal distance (c)	4
Foci	(4, 0), (-4, 0)
Vertices	(5, 0), (-5, 0)
Co-vertices	(0, 3), (0, -3)
Eccentricity (e)	0.8
Area	47.12

Summary of Ellipse Properties

What is an Ellipse Equation Calculator?

An ellipse equation calculator is a tool designed to find the standard equation of an ellipse given certain parameters, typically the coordinates of its center (h, k) and the lengths of its horizontal (rx) and vertical (ry) radii. The calculator outputs the equation in the standard form `((x-h)^2 / rx^2) + ((y-k)^2 / ry^2) = 1`, and also provides key properties like the foci, vertices, co-vertices, and eccentricity. This ellipse equation calculator is useful for students learning conic sections, engineers, and anyone needing to define or understand the geometry of an ellipse.

People who study geometry, physics (planetary orbits), or design often use an ellipse equation calculator. A common misconception is that 'a' is always the horizontal radius; in fact, 'a' is the semi-major axis, which can be horizontal or vertical depending on which radius (rx or ry) is larger.

Ellipse Equation Formula and Mathematical Explanation

The standard equation of an ellipse centered at (h, k) with horizontal radius rx and vertical radius ry is:

((x – h)² / rx²) + ((y – k)² / ry²) = 1

Where:

• (h, k) are the coordinates of the center of the ellipse.
• rx is the horizontal radius (distance from the center to the ellipse along the x-axis).
• ry is the vertical radius (distance from the center to the ellipse along the y-axis).

The semi-major axis 'a' is the larger of rx and ry, and the semi-minor axis 'b' is the smaller of rx and ry.

If rx > ry, the major axis is horizontal, a = rx, b = ry. The distance from the center to each focus 'c' is found by c² = a² – b² = rx² – ry². The foci are at (h ± c, k).

If ry > rx, the major axis is vertical, a = ry, b = rx. The distance from the center to each focus 'c' is found by c² = a² – b² = ry² – rx². The foci are at (h, k ± c).

Eccentricity (e) is c/a, and it measures how elongated the ellipse is (0 for a circle, close to 1 for a very elongated ellipse).

Variables in the Ellipse Equation and its Properties

Variable	Meaning	Unit	Typical Range
h	x-coordinate of the center	Length units	Any real number
k	y-coordinate of the center	Length units	Any real number
rx	Horizontal radius	Length units	Positive numbers
ry	Vertical radius	Length units	Positive numbers
a	Semi-major axis length (max(rx, ry))	Length units	Positive numbers
b	Semi-minor axis length (min(rx, ry))	Length units	Positive numbers
c	Distance from center to focus	Length units	0 to a
e	Eccentricity (c/a)	Dimensionless	0 to 1

Practical Examples (Real-World Use Cases)

Example 1: Satellite Orbit

A satellite orbits a planet in an elliptical path. The planet is at one focus. Let's say the center of the elliptical orbit is (0, 0), the horizontal radius is 10,000 km (rx=10000), and the vertical radius is 9,500 km (ry=9500). Using the ellipse equation calculator:

• h=0, k=0, rx=10000, ry=9500
• a=10000, b=9500, c = sqrt(10000^2 – 9500^2) ≈ 3122.5 km
• Equation: (x² / 10000²) + (y² / 9500²) = 1
• Foci: (±3122.5, 0) – The planet is at one of these points.

Example 2: Designing an Elliptical Table

An architect wants to design an elliptical table with the center at (1, 2), a width of 6 feet (so rx=3) and a length of 4 feet (so ry=2, if length is vertical). They use the ellipse equation calculator:

• h=1, k=2, rx=3, ry=2
• a=3, b=2, c = sqrt(3^2 – 2^2) = sqrt(5) ≈ 2.236
• Equation: ((x-1)² / 9) + ((y-2)² / 4) = 1
• Foci: (1 ± 2.236, 2)

How to Use This Ellipse Equation Calculator

1. Enter Center Coordinates: Input the x-coordinate (h) and y-coordinate (k) of the ellipse's center.
2. Enter Radii: Input the horizontal radius (rx) and the vertical radius (ry). These must be positive numbers.
3. Calculate: Click "Calculate" or observe the results updating automatically as you type.
4. View Equation: The primary result shows the standard equation of the ellipse.
5. Check Properties: The "Ellipse Properties" section displays the center, semi-major/minor axes, focal distance, foci coordinates, vertices, co-vertices, eccentricity, and area.
6. See Visualization: The SVG chart visually represents the ellipse, its center, and foci. The table also summarizes these properties.
7. Reset/Copy: Use "Reset" to return to default values or "Copy Results" to copy the equation and properties.

The results from the ellipse equation calculator give you the mathematical definition of the ellipse and its key geometric features.

Key Factors That Affect Ellipse Equation Results

The equation and properties of an ellipse are directly determined by:

1. Center Coordinates (h, k): Shifting the center (h, k) translates the ellipse on the coordinate plane without changing its shape or orientation. The (x-h) and (y-k) terms in the equation reflect this.
2. Horizontal Radius (rx): This determines the extent of the ellipse along the x-axis from the center. A larger rx makes the ellipse wider.
3. Vertical Radius (ry): This determines the extent of the ellipse along the y-axis from the center. A larger ry makes the ellipse taller.
4. Relative Sizes of rx and ry: If rx > ry, the major axis is horizontal. If ry > rx, the major axis is vertical. If rx = ry, the ellipse is a circle. This ratio determines the shape and orientation of the major axis.
5. Focal Distance (c): Derived from rx and ry (c² = |rx² – ry²|), 'c' determines the position of the foci. Larger differences between rx and ry lead to a larger 'c' and more distant foci from the center.
6. Eccentricity (e): Calculated as c/a (where a is the semi-major axis), eccentricity quantifies how "un-circular" the ellipse is. It's directly affected by the ratio of rx to ry.

Understanding how these inputs affect the ellipse equation calculator's output helps in predicting and interpreting the ellipse's geometry.

Frequently Asked Questions (FAQ)

What is the standard form of an ellipse equation?

The standard form is ((x-h)² / rx²) + ((y-k)² / ry²) = 1, where (h,k) is the center, rx is the horizontal radius, and ry is the vertical radius.

What if rx = ry in the ellipse equation calculator?

If rx = ry, the ellipse becomes a circle with radius rx (or ry), c=0, and eccentricity=0. The equation becomes (x-h)² + (y-k)² = rx².

How do you find the foci of an ellipse?

Calculate c = sqrt(|rx² – ry²|). If rx > ry (horizontal major axis), foci are at (h ± c, k). If ry > rx (vertical major axis), foci are at (h, k ± c). Our ellipse equation calculator does this automatically.

What does eccentricity tell me about the ellipse?

Eccentricity (e) ranges from 0 (a circle) to almost 1 (a very flat ellipse). It measures how much the ellipse deviates from being circular.

Can rx or ry be zero or negative?

No, radii must be positive values for a valid ellipse. The ellipse equation calculator will show an error for non-positive radii.

What are vertices and co-vertices?

Vertices are the endpoints of the major axis, and co-vertices are the endpoints of the minor axis.

How is the area of an ellipse calculated?

The area of an ellipse is A = π * rx * ry. The calculator provides this value.

Can I find the equation if I know the foci and major/minor axis lengths?

Yes, but this calculator requires the center and radii. If you have foci and axis lengths, you can first find the center and radii, then use this ellipse equation calculator. For other input types, you might need a different ellipse foci calculator.

Related Tools and Internal Resources

• Ellipse Foci Calculator: Find the foci of an ellipse given its equation or properties.
• Circle Equation Calculator: Calculate the equation of a circle given its center and radius.
• Parabola Equation Calculator: Find the equation of a parabola.
• Hyperbola Equation Calculator: Calculate the equation of a hyperbola.
• Graph Ellipse Equation: Learn how to graph an ellipse from its equation.
• Conic Sections Guide: An overview of ellipses, parabolas, and hyperbolas.