The Rectangular Hyperbola (Edexcel IAL Further Maths)

Equation of a Rectangular Hyperbola

Spec point — 4.1

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

  • $y=\frac{1}{x}$ 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 rectangular hyperbola

• The general equation for a rectangular hyperbola is

  • $xy=c^{2}$ in Cartesian form

  • $x=ct,y=\frac{c}{t}$ in Parametric form

    • $t\neq0$

  • where c is a positive constant

• The asymptotes are the lines $y=0$ and $x=0$

  • These are rectangular (horizontal and vertical)

    • Non-rectangular hyperbola have asymptotes at angles

• The general equation can be rearranged

  • $y=\frac{c^{2}}{x}$

    • This is a more familiar reciprocal form

Exam Hint 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 $x=ct$* *and $y=\frac{c}{t}$ where $t\neq0$ and $c>0$.

Show that its Cartesian equation is $xy=c^{2}$.

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

$t=\frac{x}{c}$ and $t=\frac{c}{y}$* *

Then set them equal to each other and cross-multiply

• $\begin{gathered} \frac{x}{c} & = \frac{c}{y} \\ x y & = c^{2} \end{gathered}$ *

$xy=c^{2}$

Other correct ways to eliminate$t$are also accepted