Summary: The Hyperbola

Key Equations

Hyperbola, center at origin, transverse axis on x-axis [latex]\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1[/latex]
Hyperbola, center at origin, transverse axis on y-axis [latex]\frac{{y}^{2}}{{a}^{2}}-\frac{{x}^{2}}{{b}^{2}}=1[/latex]
Hyperbola, center at [latex]\left(h,k\right)[/latex], transverse axis parallel to x-axis [latex]\frac{{\left(x-h\right)}^{2}}{{a}^{2}}-\frac{{\left(y-k\right)}^{2}}{{b}^{2}}=1[/latex]
Hyperbola, center at [latex]\left(h,k\right)[/latex], transverse axis parallel to y-axis [latex]\frac{{\left(y-k\right)}^{2}}{{a}^{2}}-\frac{{\left(x-h\right)}^{2}}{{b}^{2}}=1[/latex]

Key Concepts

  • A hyperbola is the set of all points [latex]\left(x,y\right)[/latex] in a plane such that the difference of the distances between [latex]\left(x,y\right)[/latex] and the foci is a positive constant.
  • The standard form of a hyperbola can be used to locate its vertices and foci.
  • When given the coordinates of the foci and vertices of a hyperbola, we can write the equation of the hyperbola in standard form.
  • When given an equation for a hyperbola, we can identify its vertices, co-vertices, foci, asymptotes, and lengths and positions of the transverse and conjugate axes in order to graph the hyperbola.
  • Real-world situations can be modeled using the standard equations of hyperbolas. For instance, given the dimensions of a natural draft cooling tower, we can find a hyperbolic equation that models its sides.

Glossary

center of a hyperbola the midpoint of both the transverse and conjugate axes of a hyperbola

conjugate axis the axis of a hyperbola that is perpendicular to the transverse axis and has the co-vertices as its endpoints

hyperbola the set of all points [latex]\left(x,y\right)[/latex] in a plane such that the difference of the distances between [latex]\left(x,y\right)[/latex] and the foci is a positive constant

transverse axis the axis of a hyperbola that includes the foci and has the vertices as its endpoints