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Analytic Geometry2D

The Hyperbola - Theory & Concepts - Analytic Geometry Hyperbola Asymptote Explorer

Equations of hyperbolas, finding foci, vertices, transverse axis, asymptotes, and eccentricity.

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Hyperbola & Asymptote Explorer

Hyperbola parameters

Semi-transverse axis a3.5
Semi-conjugate axis b2.5

Test point on hyperbola

Point P y-coordinate (y_P)2.0

Verification details

Hyperbola Standard Equation
x2a2−y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1
a=3.5, b=2.5, c=a2+b2=4.30a = 3.5, \ b = 2.5, \ c = \sqrt{a^2 + b^2} = 4.30
Asymptotes: y=±bax=±0.71xy = \pm \frac{b}{a}x = \pm 0.71x
Distances to Foci
d1=PF1=9.008, d2=PF2=2.008d_1 = PF_1 = 9.008, \ d_2 = PF_2 = 2.008
Hyperbola Locus Metric
∣d1−d2∣=∣9.008−2.008∣=7.0=2a|d_1 - d_2| = |9.008 - 2.008| = 7.0 = 2a
The focal distance difference is constant and equals the transverse axis length 7.
Focus F1 (-4.30, 0)F₁Focus F2 (4.30, 0)F₂Point P (4.5, 2.0)P(4.5, 2.0)

The asymptotes act as boundary lines (dashed gray) which the hyperbola approaches as $x$ increases

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