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

Properties of hyperbolas

What is a hyperbola?

• A hyperbola is a curve with the Cartesian equation

  • $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

  • which consists of two separate branches

    • the positive-$x$ branch ($x\ge a$)

    • the negative-$x$ branch ($x\le -a$)

  • that are bounded by the two asymptotes

    • $y=\pm \frac{b}{a}x$

  • which pass through the origin

    • and are not necessarily perpendicular

• The asymptote equations can be derived by rearranging the curve

  • $y^2=\frac{b^2{x}^2}{a^2}\left(1-\frac{a^2}{x^2}\right)$ so $y=\pm \sqrt{\frac{b^2{x}^2}{a^2}}\sqrt{1-\frac{a^2}{x^2}}$

    • as $x\to \infty$, $\frac{a^2}{x^2}\to 0$ so second square root tends to 1 and $y\to \pm \frac{b}{a}x$

• A hyperbola is one of the conic curves

  • with eccentricity $e>1$

What are the parametric equations of a hyperbola?

• There are two different sets of parametric equations for a hyperbola that are both equally valid

• The first set of parametric equations is

  • $x=\pm a\cosh \theta$

  • $y=b\sinh \theta$

  • where $\theta \in \mathrm{ℝ}$

    • and $\pm$ depends on positive or negative-$x$ branch

• The second set of parametric equations is

  • $x=a\sec \theta$

  • $y=b\tan \theta$

  • where $-\frac{\pi }{2}<\theta <\frac{\pi }{2}$ defines the positive-$x$ branch

    • as $\sec \theta$ is positive

  • and both $-\pi <\theta <-\frac{\pi }{2}$ and $\frac{\pi }{2}<\theta \le \pi$ define the negative-$x$ branch

    • as $\sec \theta$ is negative

  • and $\theta \ne \pm \frac{\pi }{2}$

    • as $\sec \theta$ and $\tan \theta$ are undefined

• Eliminating the parameter, $\theta$, gives the Cartesian equation $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

  • using either ${\cosh }^2\theta -{\sinh }^2\theta \equiv 1$

  • or $1+{\tan }^2\theta \equiv {\sec }^2\theta$ and rearranging

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

• A general point $P$ on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ has coordinates given by its parametric equations, $P\left(a\cosh \theta ,b\sinh \theta \right)$ or $P\left(a\sec \theta ,b\tan \theta \right)$

• e.g. $P\left(3\sec \theta ,2\tan \theta \right)$ is a general point on the hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$

  • It satisfies the equation of the curve

  • It moves around the curve depending on the value of $\theta$

• This is different to, say, $\left(3,0\right)$

  • which is a fixed point on the hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$

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

• The eccentricity of a hyperbola, $e$, takes the range $e>1$

• If $a>b$

  • the eccentricity is found be rearranging the following formula

    • $b^2=a^2\left(e^2-1\right)$

  • the foci, $F$ and $F\text{'}$, are two symmetric points on the x-axis enclosed by either branch

    • with coordinates $\left(\pm ae,0\right)$

  • the directrices are the two vertical lines positioned symmetrically either side of the origin

    • in the gap between the branches

    • with equations $x=\pm \frac{a}{e}$

What is the focus-directrix property of a hyperbola?

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

  • i.e. $\frac{PF}{PD}=e$

    • sometimes rearranged to $PF=ePD$

• The focus-directrix property works from $P$ to the other focus, $F\text{'}$, and directrix, $D\text{'}$

  • $\frac{PF\text{'}}{PD\text{'}}=e$

    • where $e$ is the same eccentricity