Revision NotesPast Papers
International ASFurther MathsEdexcelFurther Pure 1Revision NotesCoordinate SystemsCoordinate SystemsThe Rectangular Hyperbola

The Rectangular Hyperbola (Edexcel International AS Further Maths): Revision Note

Exam code: XFM01

Mark Curtis

Written by: Mark Curtis

Reviewed by: Dan Finlay

Updated on

The Equation of a Rectangular Hyperbola

What is a rectangular hyperbola?

  • A rectangular hyperbola is a reciprocal curve with two L-shaped branches and asymptotes along the axes

    • y equals 1 over x is a rectangular hyperbola

  • A rectangular hyperbola is part of a family of curves called the conics (or conic sections)

    • Conics are parabolae, hyperbolae and ellipses

What is the general equation of a rectangular hyperbola?

The general rectangular hyperbola
The rectangular hyperbola

  • The general equation for a rectangular hyperbola is

    • x y equals c squared in Cartesian form

    • x equals c t comma space space y equals c over t in Parametric form

      • t not equal to 0

    • where c is a positive constant

  • The asymptotes are the lines y equals 0 and x equals 0

    • These are rectangular (horizontal and vertical)

      • Non-rectangular hyperbola have asymptotes at angles

  • The general equation can be rearranged

    • y equals c squared over x

      • This is a more familiar reciprocal form

Examiner Tips and Tricks

You are given the Cartesian and parametric equations of a rectangular hyperbola in the Formulae Booklet.

Worked Example

A rectangular hyperbola has the parametric equations x equals c t and y equals c over t where t not equal to 0 and c greater than 0.

Show that its Cartesian equation is x y equals c squared.

To find the Cartesian equation, eliminate t from the parametric equations
A quick way here is to first make t the subject of both

t equals x over c and t equals c over y 

Then set them equal to each other and cross-multiply

table row cell x over c end cell equals cell c over y end cell row cell x y end cell equals cell c squared end cell end table 

x y equals c squared

Other correct ways to eliminate t are also accepted

You've read 0 of your 5 free revision notes this week

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

    Did this page help you?

    Previous:The ParabolaNext:Tangents & Normals
    Author
    Reviewer
    Mark Curtis

    Author: Mark Curtis

    Expertise: Maths Content Creator

    Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.