Properties of Rectangular Hyperbolas (Edexcel A Level Further Maths): Revision Note

Properties of rectangular hyperbolas

What is a rectangular hyperbola?

• A rectangular hyperbola is a special hyperbola with the Cartesian equation

  • 

    • where

    • e.g. the familiar reciprocal graph when

  • Its lines of symmetry are

  • Its asymptotes are the coordinate axes

    • and

  • It is rectangular because its asymptotes are perpendicular

What are the parametric equations of a rectangular hyperbola?

• The parametric equations of a rectangular hyperbola are

  • 

  • 

  • where

• Eliminating the parameter, , gives the Cartesian equation

What are the coordinates of a general point on a rectangular hyperbola?

• A general point on the rectangular hyperbola has coordinates given by its parametric equations,

• e.g. is a general point on the rectangular hyperbola  (where )

  • It satisfies the equation of the curve

  • It moves around the curve depending on the value of

• This is different to, say,

  • which is a fixed point on the rectangular hyperbola 

What is the eccentricity, focus and directrix of a rectangular hyperbola?

• The eccentricity of a rectangular hyperbola, , is

  • 

• The foci, and , are the points on the line

• The directrices are the lines with equations

  • perpendicular to the line

What is the focus-directrix property of a rectangular hyperbola?

• The focus-directrix property says that, if you take any point on a rectangular hyperbola, then

  • the distance from to the focus,

  • divided by the shortest distance from to the directrix (at point )

  • is always equal to , the eccentricity, where

  • i.e.

    • sometimes rearranged to