Find The Vertices Foci And Eccentricity Of An Ellipse Calculator

Ellipse Properties Calculator

Enter the center coordinates (h, k) and the lengths of the semi-axes along the x (rx) and y (ry) directions to find the vertices, foci, and eccentricity of the ellipse.

What is an Ellipse and its Properties?

An ellipse is a closed curve in a plane that results from the intersection of a plane with a cone in a way that the plane is not parallel or perpendicular to the cone's base and does not intersect the base. Geometrically, an ellipse is the set of all points in a plane such that the sum of the distances from two fixed points, called the foci (plural of focus), is constant. This constant sum is equal to the length of the major axis (2a).

Key properties of an ellipse include:

• Center (h, k): The midpoint of the major and minor axes.
• Major Axis: The longest diameter of the ellipse, passing through the center and both foci, with endpoints called vertices. Its length is 2a.
• Minor Axis: The shortest diameter of the ellipse, passing through the center and perpendicular to the major axis, with endpoints called co-vertices. Its length is 2b.
• Semi-major Axis (a): Half the length of the major axis.
• Semi-minor Axis (b): Half the length of the minor axis.
• Foci (F1, F2): Two fixed points on the major axis inside the ellipse used in its definition. The distance from the center to each focus is 'c'.
• Vertices: The endpoints of the major axis.
• Co-vertices: The endpoints of the minor axis.
• Eccentricity (e): A number that measures how "elongated" the ellipse is. It ranges from 0 (a circle) to just under 1 (a very flat ellipse).

This find the vertices foci and eccentricity of an ellipse calculator helps you determine these properties from the center and semi-axes lengths.

Ellipse Formula and Mathematical Explanation

For an ellipse centered at (h, k), if the semi-axis length along the x-direction is rx and along the y-direction is ry:

1. Determine 'a' and 'b': The semi-major axis 'a' is the larger of rx and ry, and the semi-minor axis 'b' is the smaller. a = max(rx, ry), b = min(rx, ry).

2. Orientation: If rx > ry, the major axis is horizontal. If ry > rx, the major axis is vertical. If rx = ry, it's a circle.

3. Distance to Foci (c): c² = a² – b², so c = √(a² – b²).

4. Eccentricity (e): e = c / a. Since c < a, 0 ≤ e < 1. For a circle, c=0, so e=0.

5. Vertices: – Horizontal major axis: (h ± a, k) – Vertical major axis: (h, k ± a)

6. Foci: – Horizontal major axis: (h ± c, k) – Vertical major axis: (h, k ± c)

7. Co-vertices: – Horizontal major axis: (h, k ± b) – Vertical major axis: (h ± b, k)

The standard equation of an ellipse centered at (h, k) is: ((x-h)²/rx²) + ((y-k)²/ry²) = 1.

Variables Table

Variable	Meaning	Unit	Typical Range
h, k	Coordinates of the center	Length units	Any real number
rx	Semi-axis length along x-direction	Length units	> 0
ry	Semi-axis length along y-direction	Length units	> 0
a	Semi-major axis length (max(rx, ry))	Length units	> 0
b	Semi-minor axis length (min(rx, ry))	Length units	> 0, b ≤ a
c	Distance from center to focus	Length units	0 ≤ c < a
e	Eccentricity	Dimensionless	0 ≤ e < 1

Table 1: Variables used in ellipse calculations.

Practical Examples

Example 1: Horizontal Ellipse

Suppose an ellipse is centered at (1, 2), with a semi-axis length of 5 along the x-direction (rx=5) and 3 along the y-direction (ry=3).

• h=1, k=2, rx=5, ry=3
• a = max(5, 3) = 5, b = min(5, 3) = 3 (Horizontal major axis)
• c = √(5² – 3²) = √(25 – 9) = √16 = 4
• e = c/a = 4/5 = 0.8
• Vertices: (1 ± 5, 2) => (6, 2) and (-4, 2)
• Foci: (1 ± 4, 2) => (5, 2) and (-3, 2)
• Co-vertices: (1, 2 ± 3) => (1, 5) and (1, -1)

Our find the vertices foci and eccentricity of an ellipse calculator would confirm these values.

Example 2: Vertical Ellipse

An ellipse is centered at (-2, 0), with rx=4 and ry=6.

• h=-2, k=0, rx=4, ry=6
• a = max(4, 6) = 6, b = min(4, 6) = 4 (Vertical major axis)
• c = √(6² – 4²) = √(36 – 16) = √20 ≈ 4.472
• e = c/a = √20 / 6 ≈ 0.745
• Vertices: (-2, 0 ± 6) => (-2, 6) and (-2, -6)
• Foci: (-2, 0 ± √20) => (-2, √20) and (-2, -√20)
• Co-vertices: (-2 ± 4, 0) => (2, 0) and (-6, 0)

How to Use This Find the Vertices Foci and Eccentricity of an Ellipse Calculator

1. Enter Center Coordinates: Input the x-coordinate (h) and y-coordinate (k) of the ellipse's center.
2. Enter Semi-axes Lengths: Input the length of the semi-axis along the x-direction (rx) and the y-direction (ry). These must be positive values.
3. Calculate: Click the "Calculate" button or simply change the input values. The results will update automatically.
4. Read Results: The calculator will display the eccentricity (e) as the primary result, along with the orientation, values of a, b, c, and the coordinates of the vertices, foci, and co-vertices.
5. Visualize: A scaled SVG graph of the ellipse with its center, foci, and vertices is shown.
6. Reset: Click "Reset" to return to default values.
7. Copy: Click "Copy Results" to copy the main findings to your clipboard.

Key Factors That Affect Ellipse Properties

• Center (h, k): Changing the center shifts the entire ellipse on the coordinate plane but does not affect its shape, eccentricity, or the lengths of a, b, and c. It only changes the absolute coordinates of vertices and foci.
• Semi-axis along x (rx): Affects the horizontal extent of the ellipse. If rx is significantly different from ry, it influences whether the major axis is horizontal or vertical and the values of a, b, c, and e.
• Semi-axis along y (ry): Affects the vertical extent of the ellipse. Similar to rx, its relative value to rx determines the orientation and properties.
• Ratio of rx to ry: The closer rx and ry are, the closer 'b' is to 'a', 'c' is to 0, and 'e' is to 0 (more circular). The more different they are, the larger 'c' and 'e' become (more elongated).
• Magnitude of rx and ry: Larger values of rx and ry mean a larger ellipse overall, affecting the absolute distances between the center, vertices, and foci, but if their ratio remains the same, the eccentricity 'e' stays the same.
• Whether rx > ry or ry > rx: This determines whether the major axis is horizontal or vertical, thus dictating the orientation of vertices and foci relative to the center.

Frequently Asked Questions (FAQ)

Q1: What is the eccentricity of a circle? A1: A circle is a special case of an ellipse where rx = ry (so a = b). In this case, c = √(a² – a²) = 0, so the eccentricity e = c/a = 0/a = 0. The two foci coincide at the center.

Q2: What does an eccentricity close to 1 mean? A2: An eccentricity close to 1 means the ellipse is very elongated or "flat". The distance 'c' to the foci is almost equal to 'a', and 'b' is very small compared to 'a'.

Q3: Can rx or ry be zero or negative? A3: No, rx and ry represent lengths (or distances from the center), so they must be positive values for a non-degenerate ellipse. Our find the vertices foci and eccentricity of an ellipse calculator requires positive inputs.

Q4: How do I know if the major axis is horizontal or vertical? A4: If rx > ry, the semi-major axis a = rx, and the major axis is horizontal. If ry > rx, the semi-major axis a = ry, and the major axis is vertical.

Q5: What are the units of eccentricity? A5: Eccentricity is a ratio of two lengths (c/a), so it is a dimensionless quantity – it has no units.

Q6: Where are the foci located relative to the vertices? A6: The foci are always located on the major axis, inside the ellipse, between the center and the vertices.

Q7: What is the latus rectum of an ellipse? A7: The latus rectum is a line segment passing through a focus, perpendicular to the major axis, with endpoints on the ellipse. Its length is 2b²/a. This calculator does not directly compute it, but it can be derived from 'a' and 'b'.

Q8: Can I use this calculator for other conic sections? A8: No, this calculator is specifically designed to find the vertices foci and eccentricity of an ellipse calculator properties. For parabolas or hyperbolas, you would need different parameters and formulas, maybe our hyperbola calculator or parabola calculator would help.