The Rectangular Hyperbola (Edexcel International A Level Further Maths)

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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

The general equation for a rectangular hyperbola is
- $x y = c^{2}$ in Cartesian form
- $x = c t, y = \frac{c}{t}$ in Parametric form
  - $t \neq 0$
- 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

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

A rectangular hyperbola has the parametric equations $x = c t$ and $y = \frac{c}{t}$ where $t \neq 0$ and $c > 0$ .

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

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

$\begin{array}{rcl} \frac{x}{c} & = & \frac{c}{y} \\ x y & = & c^{2} \end{array}$

$x y = c^{2}$

Other correct ways to eliminate $t$ are also accepted

to absolutely everything:

the (exam) results speak for themselves:

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