Enter the coordinates of the hyperbola's center, one vertex, and one focus to determine its standard form equation and other key properties.
Understanding the Standard Form of a Hyperbola
A hyperbola is a fascinating conic section defined as the set of all points in a plane where the difference of the distances from two fixed points, called the foci, is constant. This geometric property leads to two standard forms for its equation, depending on its orientation.
1. Horizontal Transverse Axis
When the hyperbola opens left and right, its transverse axis is horizontal. The standard form is:
(x - h)² / a² - (y - k)² / b² = 1
2. Vertical Transverse Axis
When the hyperbola opens up and down, its transverse axis is vertical. The standard form is:
(y - k)² / a² - (x - h)² / b² = 1
In these equations:
(h, k) represents the coordinates of the hyperbola's center.
a is the distance from the center to each vertex.
b is the distance from the center to the co-vertices, which helps define the shape and the asymptotes.
c is the distance from the center to each focus.
These values are related by the crucial formula: c² = a² + b².
How to Manually Calculate the Equation
Let's walk through an example to see how the calculator derives the equation. Suppose you are given the following points:
Center: (2, -1)
Vertex: (5, -1)
Focus: (7, -1)
Determine Orientation: The y-coordinates of the center, vertex, and focus are all -1. This means the points lie on a horizontal line, so the hyperbola has a horizontal transverse axis.
Identify (h, k): The center is given as (h, k) = (2, -1).
Calculate 'a': 'a' is the distance from the center to the vertex.
a = |Vertex x - Center x| = |5 - 2| = 3. Therefore, a² = 9.
Calculate 'c': 'c' is the distance from the center to the focus.
c = |Focus x - Center x| = |7 - 2| = 5. Therefore, c² = 25.
Calculate 'b²': Using the formula c² = a² + b², we can find b².
b² = c² - a² = 25 - 9 = 16.
Write the Equation: Substitute the values of h, k, a², and b² into the standard form for a horizontal hyperbola.
(x - 2)² / 9 - (y - (-1))² / 16 = 1 Which simplifies to: (x - 2)² / 9 - (y + 1)² / 16 = 1.
Applications of Hyperbolas
Hyperbolas are not just abstract mathematical concepts; they have significant real-world applications:
Navigation Systems: The LORAN (Long Range Navigation) system uses the time difference between signals received from two stations to place a ship or aircraft on a specific hyperbolic curve.
Astronomy: The paths of some comets that pass through our solar system only once are hyperbolic.
Optics and Telescopes: The Cassegrain telescope design uses a primary parabolic mirror and a secondary hyperbolic mirror to focus light.
Sound Ranging: By measuring the time difference at which a sound (like an explosion) is heard at multiple locations, one can determine the location of the sound source, which lies at the intersection of hyperbolas.
function calculateHyperbola() {
var h = parseFloat(document.getElementById('centerX').value);
var k = parseFloat(document.getElementById('centerY').value);
var vx = parseFloat(document.getElementById('vertexX').value);
var vy = parseFloat(document.getElementById('vertexY').value);
var fx = parseFloat(document.getElementById('focusX').value);
var fy = parseFloat(document.getElementById('focusY').value);
var resultDiv = document.getElementById('result');
if (isNaN(h) || isNaN(k) || isNaN(vx) || isNaN(vy) || isNaN(fx) || isNaN(fy)) {
resultDiv.innerHTML = 'Error: Please fill in all fields with valid numbers.';
return;
}
var a, c, b, bSquared;
var orientation = ";
var equation = ";
var asymptotes = ";
// Check for horizontal orientation
if (k === vy && k === fy && h !== vx && h !== fx) {
orientation = 'Horizontal';
a = Math.abs(vx – h);
c = Math.abs(fx – h);
}
// Check for vertical orientation
else if (h === vx && h === fx && k !== vy && k !== fy) {
orientation = 'Vertical';
a = Math.abs(vy – k);
c = Math.abs(fy – k);
}
// Invalid alignment
else {
resultDiv.innerHTML = 'Error: The center, vertex, and focus must be aligned either horizontally or vertically.';
return;
}
if (c a).';
return;
}
var aSquared = Math.pow(a, 2);
var cSquared = Math.pow(c, 2);
bSquared = cSquared – aSquared;
b = Math.sqrt(bSquared);
var xTermSign = h > 0 ? '-' : '+';
var yTermSign = k > 0 ? '-' : '+';
var absH = Math.abs(h);
var absK = Math.abs(k);
var xPart = h === 0 ? 'x²' : '(x ' + xTermSign + ' ' + absH + ')²';
var yPart = k === 0 ? 'y²' : '(y ' + yTermSign + ' ' + absK + ')²';
var yAsymptotePart = k === 0 ? 'y' : '(y ' + yTermSign + ' ' + absK + ')';
var xAsymptotePart = h === 0 ? 'x' : '(x ' + xTermSign + ' ' + absH + ')';
if (orientation === 'Horizontal') {
equation = '' + xPart + '⁄' + aSquared.toFixed(4).replace(/\.0+$/, ") + ' – ' + yPart + '⁄' + bSquared.toFixed(4).replace(/\.0+$/, ") + ' = 1′;
asymptotes = yAsymptotePart + ' = ±' + (b/a).toFixed(4).replace(/\.0+$/, ") + ' * ' + xAsymptotePart;
} else { // Vertical
equation = '' + yPart + '⁄' + aSquared.toFixed(4).replace(/\.0+$/, ") + ' – ' + xPart + '⁄' + bSquared.toFixed(4).replace(/\.0+$/, ") + ' = 1′;
asymptotes = yAsymptotePart + ' = ±' + (a/b).toFixed(4).replace(/\.0+$/, ") + ' * ' + xAsymptotePart;
}
var output = '