Conic Sections

Besides points and lines GeoGebra also provides conic sections as basic geometrical objects. To introduce hyperbolas, parabolas, ellipses and circles to your students, you could use a construction similar to the following. It dynamically visualizes the intersection of a double cone with a plane. The intersecting plane appears as a line in the construction and you can modify the slope of this plane by dragging the black point. In the bottom part of the dynamic figure you see the resulting curve of the intersection between plane and cone: a conic section.

Polar Lines

In the construction below we created an ellipse c by using the tool "Conic through five points" and its polar line p for an arbitrary point P to get the intersection points of the tangents through point P. Move the points in the figure below to transform the ellipse into other types of conic sections. In this way, you can explore the properties of a polar line for all kinds of conic sections by using a single construction.

Based on the construction above, you can now go on with the following steps:

1. Choose the tool "New Point" and click on the polar line p to create a new point F on this line.
2. Now, you can choose how to construct the polar line for F with respect to the conic section:
   • You either type the command Polar[F, c] into the Input Line and hit Enter
   • or you use "Polar line" from the toolbar, click on point F and the conic section c.

Afterwards, you should see that the polar line of point F runs through point P. The fundamental theorem of polar lines states that this is true for any position of P and F on p. You can try this out by choosing the tool "Move" from the toolbar and dragging P or F with the mouse. To investigate this mathematical phenomenon for a wider range of conic sections, simply transform the ellipse into another conic section by moving one of its five initial points.