Properties of Rectangular Hyperbolas (Edexcel A Level Further Maths: Further Pure 1): Revision Note

Properties of rectangular hyperbolas

What is a rectangular hyperbola?

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

  • $xy=c^2$

    • where $c>0$

    • e.g. the familiar reciprocal graph $y=\frac{1}{x}$ when $c=1$

  • Its lines of symmetry are $y=\pm x$

  • Its asymptotes are the coordinate axes

    • $x=0$ and $y=0$

  • 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

  • $x=ct$

  • $y=\frac{c}{t}$

  • where $t\in \mathrm{ℝ},t\ne 0$

• Eliminating the parameter, $t$, gives the Cartesian equation $xy=c^2$

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

• A general point $P$ on the rectangular hyperbola $xy=c^2$ has coordinates given by its parametric equations, $P\left(ct,\frac{c}{t}\right)$

• e.g. $P\left(3t,\frac{3}{t}\right)$ is a general point on the rectangular hyperbola $xy=9$ (where $c=3$)

  • It satisfies the equation of the curve

  • It moves around the curve depending on the value of $t$

• This is different to, say, $\left(3,3\right)$

  • which is a fixed point on the rectangular hyperbola $xy=9$

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

• The eccentricity of a rectangular hyperbola, $e$, is $\sqrt{2}$

  • $e=\sqrt{2}$

• The foci, $F$ and $F\text{'}$, are the points $\left(\pm \sqrt{2}c,\pm \sqrt{2}c\right)$ on the line $y=x$

• The directrices are the lines with equations $x+y=\pm \sqrt{2}c$

  • perpendicular to the line $y=x$

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

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

  • the distance from $P$ to the focus, $F$

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

  • is always equal to $e$, the eccentricity, where $e=\sqrt{2}$

  • i.e. $\frac{PF}{PD}=\sqrt{2}$

    • sometimes rearranged to $PF=\sqrt{2}PD$