Activities

Polar coordinates - equations of conic sections

To develop an understanding of the ellipse and hyperbola in polar form, plot the points using the tables and grids provided.

As shown below, there exists one polar equation that can be used for three types of conic section - ellipse, parabola and hyperbola.
 
Assume:
  • the pole (origin) is a focus
  • line  x = d  (d > 0) is a directrix
  • the eccentricity is e (e >0).
The equation is shown to be:

This animation assumes that d is 4. It shows the effect of  increasing the value of e.

The four main standard forms of the equation of a conic section are demonstrated below.

The values chosen for e are 0.6, 1 and 3.

In each case, the directrix is 4 units from the pole.
 




When investigating the shape of a conic section, it is useful to consider coordinates for particular values of .
 
In the case of an ellipse or hyperbola:

In the case of a parabola:

Calculate these coordinates for d = 4 and e = 0.6, 1 and 3. Check your results on the following diagram.
 
 

Find the eccentricity, equation of the curve and equation of the directrix.