Find Equation Of Ellipse Given Foci And Vertices Calculator

Equation of Ellipse Given Foci and Vertices Calculator

Equation of Ellipse Given Foci and Vertices Calculator

Ellipse Equation Calculator

Enter the coordinates of the two foci and two vertices (on the major axis) to find the equation of the ellipse.

Focus 1 (F1):
Focus 2 (F2):
Vertex 1 (V1):
Vertex 2 (V2):
Enter valid coordinates above.
C(0,0) F1(-3,0) F2(3,0) V1(-5,0) V2(5,0)

Visual representation of the ellipse, foci, and vertices.

Parameter Symbol Value
Center (h, k)(h, k)
Semi-major axisa
Semi-minor axisb
Focal distance from centerc
a²a²
b²b²
Orientation–

What is an Equation of Ellipse Given Foci and Vertices Calculator?

An equation of ellipse given foci and vertices calculator is a tool used to determine the standard equation of an ellipse when you know the coordinates of its two foci (F1 and F2) and its two major vertices (V1 and V2). This calculator simplifies the process of finding the center (h, k), the lengths of the semi-major (a) and semi-minor (b) axes, and the orientation of the ellipse, ultimately providing the equation in the standard form:
(x-h)²/a² + (y-k)²/b² = 1 (for a horizontal major axis) or
(x-h)²/b² + (y-k)²/a² = 1 (for a vertical major axis).

This tool is useful for students studying conic sections in algebra or pre-calculus, engineers, physicists, and anyone working with elliptical shapes. It eliminates manual calculations, which can be prone to errors, especially when determining the values of a, b, and c, and the center coordinates.

Common misconceptions include thinking that any four points define an ellipse (you need specific points like foci and vertices) or that the foci are the same as vertices (they are distinct, with foci inside the vertices on the major axis).

Equation of Ellipse Given Foci and Vertices Formula and Mathematical Explanation

An ellipse is defined as the set of all points in a plane such that the sum of the distances from two fixed points (the foci) is constant (equal to 2a, the length of the major axis).

Given foci F1(x1, y1), F2(x2, y2) and major vertices V1(vx1, vy1), V2(vx2, vy2):

  1. Find the Center (h, k): The center of the ellipse is the midpoint of the segment connecting the two foci, and also the midpoint of the segment connecting the two vertices.
    h = (x1 + x2) / 2 = (vx1 + vx2) / 2
    k = (y1 + y2) / 2 = (vy1 + vy2) / 2
  2. Find the distance 2c (between foci) and c:
    2c = sqrt((x2 - x1)² + (y2 - y1)²)
    c = 2c / 2
  3. Find the distance 2a (between vertices – major axis length) and a:
    2a = sqrt((vx2 - vx1)² + (vy2 - vy1)²)
    a = 2a / 2
  4. Find b² and b: From the relationship a² = b² + c², we get b² = a² – c². For an ellipse, a must be greater than c.
    b² = a² - c²
    b = sqrt(a² - c²) (b > 0)
  5. Determine Orientation:
    • If y1 = y2 (and vy1 = vy2), the major axis is horizontal. The equation is: (x-h)²/a² + (y-k)²/b² = 1.
    • If x1 = x2 (and vx1 = vx2), the major axis is vertical. The equation is: (x-h)²/b² + (y-k)²/a² = 1.

The equation of ellipse given foci and vertices calculator automates these steps.

Variables Table

Variable Meaning Unit Typical range
(x1, y1), (x2, y2) Coordinates of the foci Length units Real numbers
(vx1, vy1), (vx2, vy2) Coordinates of the major vertices Length units Real numbers
(h, k) Coordinates of the center Length units Real numbers
a Length of the semi-major axis Length units a > 0
b Length of the semi-minor axis Length units b > 0
c Distance from the center to each focus Length units 0 ≤ c < a

Practical Examples

Example 1: Horizontal Ellipse

Suppose the foci are F1(-4, 2) and F2(4, 2), and the vertices are V1(-5, 2) and V2(5, 2).

  • Center: h = (-4+4)/2 = 0, k = (2+2)/2 = 2. So, (h, k) = (0, 2).
  • 2c = sqrt((4 – (-4))² + (2 – 2)²) = sqrt(8² + 0²) = 8, so c = 4.
  • 2a = sqrt((5 – (-5))² + (2 – 2)²) = sqrt(10² + 0²) = 10, so a = 5.
  • b² = a² – c² = 5² – 4² = 25 – 16 = 9, so b = 3.
  • Since the y-coordinates of foci and vertices are the same, the major axis is horizontal.
  • Equation: (x – 0)²/5² + (y – 2)²/3² = 1 => x²/25 + (y – 2)²/9 = 1

The equation of ellipse given foci and vertices calculator would confirm this result.

Example 2: Vertical Ellipse

Suppose the foci are F1(1, -2) and F2(1, 4), and the vertices are V1(1, -3) and V2(1, 5).

  • Center: h = (1+1)/2 = 1, k = (-2+4)/2 = 1. So, (h, k) = (1, 1).
  • 2c = sqrt((1 – 1)² + (4 – (-2))²) = sqrt(0² + 6²) = 6, so c = 3.
  • 2a = sqrt((1 – 1)² + (5 – (-3))²) = sqrt(0² + 8²) = 8, so a = 4.
  • b² = a² – c² = 4² – 3² = 16 – 9 = 7, so b = sqrt(7) ≈ 2.646.
  • Since the x-coordinates of foci and vertices are the same, the major axis is vertical.
  • Equation: (x – 1)²/(sqrt(7))² + (y – 1)²/4² = 1 => (x – 1)²/7 + (y – 1)²/16 = 1

How to Use This Equation of Ellipse Given Foci and Vertices Calculator

  1. Enter Foci Coordinates: Input the x and y coordinates for both Focus 1 (F1) and Focus 2 (F2).
  2. Enter Vertices Coordinates: Input the x and y coordinates for both Vertex 1 (V1) and Vertex 2 (V2). These must be the vertices along the major axis.
  3. Check Results: The calculator will instantly display:
    • The standard equation of the ellipse.
    • The coordinates of the center (h, k).
    • The values of a, b, and c.
    • The orientation (horizontal or vertical major axis).
    • A visual representation of the ellipse.
    • A table summarizing the parameters.
  4. Error Handling: If the provided points do not form a valid ellipse (e.g., if a is not greater than c, or if the points are not collinear in a way that suggests a horizontal or vertical major axis with the center being the midpoint), an error message will guide you.

Using the equation of ellipse given foci and vertices calculator helps visualize and understand the properties of the ellipse derived from these key points.

Key Factors That Affect Equation of Ellipse Given Foci and Vertices Calculator Results

  1. Coordinates of Foci: The distance between the foci (2c) directly influences the 'flatness' of the ellipse. The midpoint of the foci determines the center.
  2. Coordinates of Vertices: The distance between the vertices (2a) defines the length of the major axis. The midpoint also determines the center.
  3. Collinearity and Midpoint Consistency: The foci and vertices must lie on the same line (the major axis), and both pairs must share the same midpoint (the center of the ellipse). Our equation of ellipse given foci and vertices calculator checks for this implicitly.
  4. Relative Distance (a vs c): The distance 'a' (from center to vertex) must be greater than 'c' (from center to focus). If a ≤ c, the points do not define an ellipse in the standard way.
  5. Orientation: Whether the foci and vertices lie on a horizontal or vertical line determines if the major axis is horizontal or vertical, thus changing which denominator (a² or b²) goes under the (x-h)² term.
  6. Input Precision: The accuracy of the input coordinates will directly affect the calculated values of h, k, a, b, c, and the final equation. Small changes in input can alter the ellipse's shape and position.

Frequently Asked Questions (FAQ)

1. What if the foci and vertices do not lie on a horizontal or vertical line?

If the foci and vertices define a major axis that is tilted, the ellipse equation is more complex, involving an xy-term. This calculator is designed for ellipses with horizontal or vertical major axes.

2. What if the distance between vertices (2a) is less than or equal to the distance between foci (2c)?

If 2a ≤ 2c (or a ≤ c), then b² = a² – c² would be zero or negative, which is not possible for a real ellipse. It means the given points don't form an ellipse with those foci and major vertices.

3. Can I enter the coordinates of the minor axis vertices instead?

This specific equation of ellipse given foci and vertices calculator requires the major axis vertices because their distance directly gives 2a. If you have minor axis vertices, you'd need different information (like 'a' or 'c') to find the equation.

4. How is 'b' calculated?

'b' is the length of the semi-minor axis and is found using the relationship a² = b² + c², so b = sqrt(a² – c²).

5. What does the value 'c' represent?

'c' is the distance from the center of the ellipse to each focus.

6. What if I only have one focus and one vertex?

You need both foci and both major vertices (or equivalent information like the center and 'a' and 'c') to uniquely define the ellipse's equation using this method.

7. Does the order of entering F1, F2 or V1, V2 matter?

No, the order does not matter as the calculator calculates distances and midpoints, which are independent of order.

8. Can 'a' or 'b' be negative?

No, 'a' and 'b' represent lengths (semi-major and semi-minor axes), so they are always positive.

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