What is a Hyperbola?

A hyperbola is a type of curve that forms when a double cone intersects with a plane. Unlike other conic sections like circles or ellipses, a hyperbola has two separate branches. These branches are mirror images of each other and are characterised by their symmetry around the hyperbola's centre.

The general equation of a hyperbola is:

Horizontal Hyperbola: \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \)
Vertical Hyperbola: \( \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 \)

Here:

• \( (h, k) \) represents the centre of the hyperbola.
• \( a \) is the distance from the centre to the vertices (along the transverse axis).
• \( b \) is the distance from the centre to the co-vertices (along the conjugate axis).

About the Hyperbola Calculator

The Hyperbola Calculator is a tool that assists you in solving and visualising hyperbolas based on their equations. Whether you're studying conic sections or need a quick tool for graphing and analysis, this calculator makes your work easier by providing accurate solutions and graphs for both horizontal and vertical hyperbolas.

Key Features

• Predefined Examples: Select from built-in examples of both horizontal and vertical hyperbolas.
• Custom Equations: Enter your own hyperbola equations for calculations.
• Dynamic Visualisation: Graphs are automatically generated to show the hyperbola.
• Key Parameters: Instantly see values like the centre, vertices, foci, and lengths of the axes.
• Step-by-Step Solutions: Detailed steps explain how each calculation is carried out.

How to Use the Hyperbola Calculator

1. Select an Example: Use the dropdown to pick a preloaded example of a horizontal or vertical hyperbola.
2. Enter a Custom Equation: Alternatively, input your own hyperbola equation in standard form (e.g., \( x^2/9 - y^2/16 = 1 \)).
3. View Results: Click on the Calculate button to see key points, such as:
   • Centre
   • Vertices
   • Foci
   • Transverse and Conjugate Axis Lengths
4. Graph the Hyperbola: The calculator will show the graph of the hyperbola, including its asymptotes.
5. Clear: Use the Clear button to reset the calculator and start again.

Understanding the Results

After calculating the hyperbola, the following key elements will be displayed:

• Centre (\( h, k \)): The midpoint of the hyperbola's symmetry.
• Vertices: Points on the transverse axis at a distance \( a \) from the centre.
• Co-vertices: Points on the conjugate axis at a distance \( b \) from the centre.
• Foci: Points located at a distance \( c \) from the centre, where \( c = \sqrt{a^2 + b^2} \).
• Asymptotes: Straight lines that the hyperbola approaches but never touches.

Graph Visualisation

The calculator creates an interactive graph of the hyperbola, which includes:

• The branches of the hyperbola.
• Asymptotes for reference.
• Key points such as vertices, co-vertices, and foci.

This visual aid helps you understand how the hyperbola behaves and how its key components relate to the equation.

Frequently Asked Questions (FAQ)

What is the difference between a horizontal and a vertical hyperbola?

In a horizontal hyperbola, the transverse axis runs horizontally, and the equation is \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \). In a vertical hyperbola, the transverse axis runs vertically, and the equation is \( \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 \).

What are asymptotes in a hyperbola?

Asymptotes are straight lines that the hyperbola gets closer to as the branches extend infinitely. For a horizontal hyperbola, the asymptotes are \( y = \pm \frac{b}{a}(x-h) + k \), and for a vertical hyperbola, they are \( y = \pm \frac{a}{b}(x-h) + k \).

How do I find the foci of a hyperbola?

The foci are found at a distance \( c \) from the centre, where \( c = \sqrt{a^2 + b^2} \). For a horizontal hyperbola, the foci are at \( (h-c, k) \) and \( (h+c, k) \). For a vertical hyperbola, they are at \( (h, k-c) \) and \( (h, k+c) \).

Can I input a custom equation?

Yes, you can enter your own hyperbola equation in standard form. The calculator will interpret the equation, identify key components, and generate the results and graph for you.

Why Use the Hyperbola Calculator?

This tool makes it easier to analyse hyperbolas by automating complex calculations and providing clear, visual results. Whether you're a student, teacher, or professional, the Hyperbola Calculator saves time and ensures accuracy when working with hyperbolas.