Step By Step Calculus » 14.3 - Conic Sections in Polar Co-ordinates

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Synopsis
The conic sections, i.e. ellipse, circle, parabola, and hyperbola, can also be defined in the following way.
“A conic section is a curve traced out by a point PP that moves in the plane of a fixed point FF (called the focus) and a fixed line ll (called the directrix) with FF not on ll such that the ratio of the distance of PP from FF to its distance from ll is a positive constant ee called the eccentricity. If 0<e<1,0<e<1, the conic is an ellipse, if e=1,e=1, the conic is a parabola, and if e>1,e>1, it is a hyperbola.”
Using this definition, one can obtain a polar equation to represent all types of conic sections. Considering pole as one focus of a conic section, if the directrix is parallel to the line \theta=\pi/2\theta=\pi/2 and located dd distance away on the left side from the pole, then the polar equation is \displaystyle r=\frac{ed}{1-e\cos\theta}.\displaystyle r=\frac{ed}{1-e\cos\theta}. If the directrix is on the right side of pole, then the polar equation is \displaystyle r=\frac{ed}{1+e\cos\theta}.\displaystyle r=\frac{ed}{1+e\cos\theta}. Similarly, if the directrix is parallel to the polar axis, then the equation becomes \displaystyle r=\frac{ed}{1\pm e\sin\theta}.\displaystyle r=\frac{ed}{1\pm e\sin\theta}.