# Polar Plots of Conic Sections

Conic sections (the circle, ellipse, parabola and hyperbola) can all be represented by an equation in polar coordinates: ,
where is the semilatus rectum and is the eccentricity of the curve. For the circle, ellipse, parabola and hyperbola, the eccentricity has the values , , and , respectively.
One focus of such a conic is at the origin. After selecting values of and , you can use the "polar angle" slider to trace out the polar curve by sweeping from 0 to . For , the plot generates both branches of a hyperbola.

### DETAILS

A more general form for the polar equation is ,
where one focus is at the left-hand side of the conic for the minus sign (which we have used) but on the right-hand side of the conic for the plus sign. The alternative form rotates the conics by 90°, with the semimajor axis oriented vertically.
The following illustration shows the derivation of the formula for the case of a parabola ( ), with the focus at and the directrix (the vertical line) on the left. At every point on the curve, the defining relation is or .
The semilatus rectum is equal to when . We obtain thereby .

### PERMANENT CITATION

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