Hyperbola Calculator
Solve horizontal and vertical hyperbolas for standard form, center $(h, k)$, transverse axis ($a$), conjugate axis ($b$), linear asymptotes, foci coordinates, and eccentricity ($e > 1$) with interactive SVG plots.
Hyperbola Definition & Standard Formulas
A hyperbola is the open conic section with eccentricity e > 1, defined as the locus of points where the absolute difference in distances to two foci is constant (2a). Its open branches are bounded by two linear asymptotes passing through the center.
Hyperbola Conic Definition & Two-Foci Distance Difference Locus
A hyperbola is the geometric locus of all points $P(x, y)$ in a plane such that the absolute difference of its distances from two fixed points $F_1$ and $F_2$ (the foci) is constant and equal to the transverse axis length $2a$:
The line segment of length $2a$ connecting the vertices through the center and foci is the transverse axis. The perpendicular line segment of length $2b$ through the center is the conjugate axis.
Standard Form Equations (Horizontal vs Vertical Transverse Axis)
- Transverse axis is horizontal ($y = k$).
- Vertices: $(h \pm a, k)$
- Foci: $(h \pm c, k)$ where $c = \sqrt{a^2 + b^2}$
- Asymptotes: $y - k = \pm \frac{b}{a}(x - h)$
- Transverse axis is vertical ($x = h$).
- Vertices: $(h, k \pm a)$
- Foci: $(h, k \pm c)$ where $c = \sqrt{a^2 + b^2}$
- Asymptotes: $y - k = \pm \frac{a}{b}(x - h)$
Linear Asymptotes Equations & Central Rectangle Geometry
The asymptotes of a hyperbola act as guiding linear boundaries that the curve approaches infinitely closely as $|x| \to \infty$. They are found by replacing the right-hand constant 1 with 0:
Geometrically, the asymptotes pass through the diagonals of the central bounding rectangle of width $2a$ and height $2b$ centered at $(h, k)$.
Focal Parameter ($c = \sqrt{a^2 + b^2}$) & Eccentricity ($e > 1$)
For a hyperbola, $c > a$ always, meaning the foci lie further from the center than the vertices.
Measures the openness of the hyperbolic branches. Larger $e$ produces wider asymptote angles.
Real-World Applications (LORAN Navigation & Gravitational Slingshots)
LORAN Radio Navigation
Coastal navigation receivers measure constant arrival time differences $\Delta t$ between transmitter pairs ($F_1, F_2$), tracing hyperbolic position lines on marine charts.
Gravitational Slingshots
Spacecraft (like Voyager 1 & 2) performing planetary flybys exceed escape velocity, traveling on open hyperbolic trajectories ($e > 1$) relative to the planet.
Nuclear Cooling Towers
Hyperboloid structural profiles minimize concrete volume while maximizing upward draft airflow and structural rigidity against high crosswinds.
Graded Step-by-Step Numerical Solutions
Analyze the hyperbola $\frac{(x - 1)^2}{9} - \frac{(y + 2)^2}{16} = 1$.
1. Center $(h, k) = (1, -2)$.
2. Positive term is $x$, so transverse axis is horizontal ($y = -2$). $a^2 = 9 \implies a = 3$, $b^2 = 16 \implies b = 4$.
3. Vertices: $(1 \pm 3, -2) \implies (4, -2)$ and $(-2, -2)$.
4. Focal distance $c = \sqrt{9 + 16} = \sqrt{25} = 5$. Foci: $(1 \pm 5, -2) \implies F_1(6, -2), F_2(-4, -2)$.
5. Asymptotes: $y - (-2) = \pm \frac{4}{3}(x - 1) \implies y + 2 = \pm \frac{4}{3}(x - 1)$.
Eccentricity $e = 5/3 \approx 1.667$.
Common Pitfalls & Minus Sign Orientation Rules
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