## Saturday, June 25, 2011

### Hyperbola

A hyperbola which is centered at (h,k) has a standard equation

$\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$ ------(1)

if the hyperbola is opening sidewards, else, if it's opeing upwards or downwards, the standard equation will be

$\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1$ ------(2)

Let's take a look at equation type (1):

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

This hyperbola in blue has
a) vertices at B1 and B2 at $(h\pm a,k)$,
b) foci at F1 and F2 at $(h\pm c,k)$ where $c=\sqrt{a^2+b^2}$,
c) asymptotes in red dotted lines $y-k=\pm \frac{b}{a}(x-h)$.

Now, let's look at equation type (2):

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

This hyperbola in blue has
a) vertices at B1 and B2 at $(h,k\pm a)$,
b) foci at F1 and F2 at $(h,k\pm c)$ where $c=\sqrt{a^2+b^2}$,
c) asymptotes in red dotted lines $y-k=\pm \frac{a}{b}(x-h)$.

Betty, Created with GeoGebra