This worksheet shows another way to construct a hyperbola or ellipse—this time using three parameters: vector A (blue), vector B (blue) and angle θ (the angle between Vector T (also blue) and the x-axis). Also shown is how to construct the curves' foci and directrixes. The parameters and plotted points for the constructions are the same as those used in and returned by the standard parametric equations for each curve.
The parametric equation for an ellipse is:
x = a * cos(θ) y = b * sin(θ)The parametric formula for a hyperbola is:
x = a * sec(θ) y = b * tan(θ)Simply enter the lengths of vectors A and B, as well as the value for θ, into the equations above to achieve the same plotted points, P1 and P2.
Drag vectors A and B to change the shape of the curve(s).
Drag vector T to trace the loci of points P1 and P2.
The blue circle is the circle of unit radius, and is used to calculate the value of θ, only.
Check the "Show components" and "Show construction" options for each curve to see the curve components and constructions.
Click here to download the worksheet. Note: you do not need to extract the contents of the Zip file. Simply change the extension from ".zip" to ".ggb" and it will work. You can download an old Geometer's Sketchpad version of this worksheet, here.
Created by Michael Horvath using GeoGebra. Last modified: Monday, 08-Dec-08 13:27:15 PST