MATHEMATICS LAB · GEOMETRY

Study of the Hyperbola

Explore both branches, asymptotes, foci, and directrices. Change the center and semiaxes to examine intersections and tangents with guided calculations.

Real, nondegenerate, axis-aligned hyperbolas are supported (not rotated). The semiaxes a e b must be positive and do not need to be ordered.

1. Define the hyperbola

Center, semiaxes, or coefficients

Equation formTransverse axis

Examples:

Display options

Controls and information

Grid Foci and rectangle Asymptotes Directrices

Drag the center O to move the hyperbola; drag V and the handle b to change the semiaxes. When tangents are enabled, drag point T along a branch.

Hyperbola and vertices Asymptotes Foci Directrices Line and intersections Tangent and T

3. Additional options

Line and tangent appear on the graph

Intersection with line ℓLine equation

Tangent at point T on the hyperbolaBranchHyperbolic parameter t Also show the normal at T

The parameter t is a real number, not an angle. A single intersection does not always indicate tangency: a line parallel to an asymptote can also meet the hyperbola once.

2. Interactive hyperbola graph

Axes at 1:1 scale

Two branches, asymptotes and foci

4. Results

Updated values

5. Geometric construction

Asymptotes, foci and focal property

6. Explained calculations

Formulas and numerical substitutions

What you can study in this version

Hyperbola and asymptotes

Center, vertices, semiaxes, foci, eccentricity, directrices and asymptotes, including shifted hyperbolas.

Intersections

Secant, tangent, external, or asymptote-parallel lines, with explained solutions.

Tangents and normals

Point T on either branch, with tangent and normal, updated graph and explained calculations.

Rotated and degenerate hyperbolas are excluded. Lines are calculated using double-precision arithmetic; near-tangent cases require care.