Find Equation Of Hyperbola Given Foci And Vertices Calculator

Find Equation of Hyperbola Given Foci and Vertices Calculator

Find Equation of Hyperbola Given Foci and Vertices Calculator

Enter the coordinates of the two foci and two vertices to find the equation of the hyperbola.

Enter values and click Calculate.

What is a Find Equation of Hyperbola Given Foci and Vertices Calculator?

A "find equation of hyperbola given foci and vertices calculator" is a tool used to determine the standard equation of a hyperbola when you know the coordinates of its two foci and two vertices. A hyperbola is a type of conic section defined as the set of all points in a plane, the difference of whose distances from two fixed points (the foci) is a constant. The vertices are the points where the hyperbola intersects its transverse axis (the line segment connecting the vertices and containing the foci).

This calculator helps students, engineers, and mathematicians quickly find the equation, center, and other key parameters (a², b², c²) of the hyperbola without manual calculations. It's particularly useful for verifying homework or for quick checks in design and analysis problems involving hyperbolic shapes.

Common misconceptions include thinking any four points define a hyperbola or that foci and vertices can be placed arbitrarily. Foci and vertices must lie on the same axis (the transverse axis), and the vertices must be between the foci.

Find Equation of Hyperbola Given Foci and Vertices Calculator: Formula and Mathematical Explanation

Given the foci F1(x1, y1), F2(x2, y2) and vertices V1(x3, y3), V2(x4, y4) of a hyperbola:

  1. Find the Center (h, k): The center of the hyperbola is the midpoint of the segment connecting the foci, and also the midpoint of the segment connecting the vertices.
    h = (x1 + x2) / 2 = (x3 + x4) / 2
    k = (y1 + y2) / 2 = (y3 + y4) / 2
  2. Determine the distance 'c' from the center to a focus:
    c = √((x1 – h)² + (y1 – k)²) = √((x2 – h)² + (y2 – k)²)
    c² = (x1 – h)² + (y1 – k)²
  3. Determine the distance 'a' from the center to a vertex:
    a = √((x3 – h)² + (y3 – k)²) = √((x4 – h)² + (y4 – k)²)
    a² = (x3 – h)² + (y3 – k)²
  4. Calculate b²: For a hyperbola, the relationship between a, b, and c is c² = a² + b². Therefore, b² = c² – a².
  5. Determine the Orientation:
    • If the y-coordinates of the foci and vertices are the same (y1 = y2 = y3 = y4 = k), the transverse axis is horizontal, and the equation is:
      (x – h)² / a² – (y – k)² / b² = 1
    • If the x-coordinates of the foci and vertices are the same (x1 = x2 = x3 = x4 = h), the transverse axis is vertical, and the equation is:
      (y – k)² / a² – (x – h)² / b² = 1

The find equation of hyperbola given foci and vertices calculator uses these steps.

Variables Used
Variable Meaning Unit Typical range
(x1, y1), (x2, y2) Coordinates of the foci Length units Real numbers
(x3, y3), (x4, y4) Coordinates of the vertices Length units Real numbers
(h, k) Coordinates of the center Length units Real numbers
a Distance from center to a vertex Length units Positive real numbers
c Distance from center to a focus Length units Positive real numbers (c > a)
b Related to the conjugate axis (b² = c² – a²) Length units Positive real numbers
a², b², c² Squares of a, b, c Squared length units Positive real numbers

Practical Examples (Real-World Use Cases)

Example 1: Horizontal Transverse Axis

Suppose the foci are at (-10, 2) and (10, 2), and the vertices are at (-6, 2) and (6, 2).

  1. Center (h, k) = ((-10+10)/2, (2+2)/2) = (0, 2)
  2. c = distance from (0, 2) to (10, 2) = 10, so c² = 100
  3. a = distance from (0, 2) to (6, 2) = 6, so a² = 36
  4. b² = c² – a² = 100 – 36 = 64
  5. Since y-coordinates are the same, it's horizontal: (x – 0)² / 36 – (y – 2)² / 64 = 1
    Equation: x²/36 – (y – 2)²/64 = 1

The find equation of hyperbola given foci and vertices calculator would confirm this.

Example 2: Vertical Transverse Axis

Suppose the foci are at (3, 8) and (3, -8), and the vertices are at (3, 5) and (3, -5).

  1. Center (h, k) = ((3+3)/2, (8-8)/2) = (3, 0)
  2. c = distance from (3, 0) to (3, 8) = 8, so c² = 64
  3. a = distance from (3, 0) to (3, 5) = 5, so a² = 25
  4. b² = c² – a² = 64 – 25 = 39
  5. Since x-coordinates are the same, it's vertical: (y – 0)² / 25 – (x – 3)² / 39 = 1
    Equation: y²/25 – (x – 3)²/39 = 1

Our find equation of hyperbola given foci and vertices calculator can solve this too.

How to Use This Find Equation of Hyperbola Given Foci and Vertices Calculator

  1. Enter Foci Coordinates: Input the x and y coordinates for the first focus (F1x, F1y) and the second focus (F2x, F2y).
  2. Enter Vertices Coordinates: Input the x and y coordinates for the first vertex (V1x, V1y) and the second vertex (V2x, V2y).
  3. Calculate: The calculator automatically updates, or you can click "Calculate".
  4. Review Results: The calculator will display:
    • The equation of the hyperbola in standard form.
    • The coordinates of the center (h, k).
    • The values of a², b², and c².
    • The orientation (horizontal or vertical transverse axis).
    • A bar chart visualizing a², b², and c².
    • A table summarizing the hyperbola's properties.
  5. Reset: Click "Reset" to clear the fields to default values.
  6. Copy: Click "Copy Results" to copy the main equation and parameters to your clipboard.

Ensure that the foci and vertices lie on the same horizontal or vertical line and that the vertices are between the foci. The find equation of hyperbola given foci and vertices calculator validates these conditions.

Key Factors That Affect Hyperbola Equation Results

  1. Coordinates of Foci (F1, F2): These directly determine the center of the hyperbola and the value of 'c'. Changing the foci coordinates shifts the hyperbola and changes 'c'.
  2. Coordinates of Vertices (V1, V2): These also determine the center and the value of 'a'. The distance between vertices is 2a. Changing vertices' positions changes 'a'.
  3. Relative Position of Foci and Vertices: The line passing through foci and vertices determines the transverse axis. If they share the same y-coordinate, the axis is horizontal; if they share the x-coordinate, it's vertical. This dictates the form of the equation.
  4. Distance between Foci (2c): A larger distance between foci (for a fixed 'a') means a larger 'c', leading to a larger 'b²' and a more "open" hyperbola along the conjugate axis direction.
  5. Distance between Vertices (2a): A larger distance between vertices means a larger 'a'. If 'c' is fixed, a larger 'a' means a smaller 'b²'.
  6. Center (h, k): The midpoint of both the foci and the vertices determines the center, which shifts the entire hyperbola on the coordinate plane, affecting the (x-h) and (y-k) terms in the equation.

Using a find equation of hyperbola given foci and vertices calculator helps visualize how these factors interact.

Frequently Asked Questions (FAQ)

Q: What if the foci and vertices do not lie on the same horizontal or vertical line? A: The standard form of the hyperbola equation (that this calculator uses) assumes the transverse axis is either horizontal or vertical. If they don't, the hyperbola is rotated, and its equation is more complex, involving an xy-term. This calculator is for non-rotated hyperbolas.
Q: Can the vertices be outside the foci? A: No, for a hyperbola, the vertices always lie on the segment between the two foci. The distance 'a' (center to vertex) is always less than 'c' (center to focus).
Q: What does b² represent? A: b² is related to the conjugate axis of the hyperbola. The length of the conjugate axis is 2b. It helps define the asymptotes of the hyperbola.
Q: How does the find equation of hyperbola given foci and vertices calculator determine the orientation? A: It compares the x and y coordinates of the foci and vertices. If the y-coordinates are the same, it's horizontal. If the x-coordinates are the same, it's vertical. It also checks if the center derived from foci matches the center from vertices.
Q: Can a or b be zero? A: No, for a hyperbola, both 'a' and 'b' (and thus a² and b²) must be positive. If b² were zero, c² would equal a², which is not possible for distinct foci and vertices of a hyperbola.
Q: What are the asymptotes of the hyperbola? A: For a horizontal hyperbola, asymptotes are y – k = ±(b/a)(x – h). For a vertical hyperbola, y – k = ±(a/b)(x – h). The calculator gives you a² and b², so you can find a and b to get the asymptotes.
Q: Why use a find equation of hyperbola given foci and vertices calculator? A: It saves time, reduces calculation errors, and provides immediate results, including the center and a², b², c², along with the standard equation.
Q: What if I enter the same point for both foci or both vertices? A: The calculator will show an error because foci and vertices must be distinct points, and c and a must be greater than zero.

Related Tools and Internal Resources

These tools can help you further explore conic sections and their properties. The find equation of hyperbola given foci and vertices calculator is one of many useful geometry tools.

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