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Hyperbola (x/a)² − (y/b)² = 1

Conic Sections · Class XI

Two branches and oblique asymptotes , adjust a, b to see the shape stretch.

2
type a value
0.56
2
type a value
0.56
Live values
  • Eccentricity1.4142
  • Focal distance2.8284
  • Asymptote slope1
xy
  • Upper branch y = b √((x/a)² − 1)
  • Lower branch y = −b √((x/a)² − 1)

Formulas in this lab

  • Eccentricity
    e=1+b2/a2e = \sqrt{1 + b^2/a^2}
  • Focal distance
    c=a2+b2c = \sqrt{a^2 + b^2}
  • Asymptote slope
    ±b/a\pm b/a
Tip: For |x| < a there are no real y values , that's the gap between the branches.

Frequently asked questions

▶What is the standard equation of a hyperbola?

The hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ has two branches opening left and right. Its eccentricity is $e = \sqrt{1 + b^2/a^2}$, with $e > 1$. The asymptotes are the lines $y = \pm(b/a)x$.

▶How do I use the hyperbola explorer lab?

Slide $a$ and $b$. The lab plots both branches plus the asymptotes and reports the foci at $(\pm c, 0)$ with $c = \sqrt{a^2 + b^2}$. Try $a = 3$, $b = 4$: $c = 5$ and $e = 5/3$.

▶Hyperbola versus ellipse: what is the sign difference doing?

Ellipse has $+$ between the terms: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, giving a closed curve. Hyperbola has $-$, giving two unbounded branches. Eccentricity formulas differ: ellipse has $c^2 = a^2 - b^2$, hyperbola has $c^2 = a^2 + b^2$. Mixing these signs is a classic JEE trap.

▶Where do hyperbolas appear in real life?

The cooling-tower shadows at thermal power plants are hyperbolic. Trajectories of comets escaping the solar system, and GPS time-difference of arrival (TDOA) hyperbolas used for location fixing in your phone, also rely on this curve. The lab's asymptote lines preview that long-range behaviour.