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Compute some properties of a conic section and find a polar equation

by RoRi

Consider a conic section with focus at the origin, eccentricity e and directrix given by

    \[ e = 1; \quad \text{directrix}: 4x + 3y = 25. \]

Compute the distance d from the focus to the directrix and find a polar equation for C. For a hyperbola, give a polar equation for each branch.

First, the distance from any point X to the directrix L is given by the equation

    \[ d(X,L) = \frac{|(X-P) \cdot N|}{\lVert N \rVert} \]

where P is any point on the directrix L. In this case we can choose P = \left( \frac{25}{4}, 0\right) as a point on the directrix and let X = (0,0) be the focus. Then the distance from the focus to the directrix is

    \[ d = \frac{P \cdot N}{\lVert N \rVert}. \]

Since a normal to the directrix is N = (4,3) (from the Cartesian equation for the directrix) we have

    \[ d = \frac{ \left( \frac{25}{4}, 0 \right) \cdot (4,3) }{5} = 5. \]

To obtain the polar equation for the conic section, since the focus is at the origin, we have the conic section is the set of points X such that

    \[ \lVert X \rVert = e| X \cdot N - d|. \]

Letting X have polar coordinates r and \theta, (i.e., X = (r \cos \theta, r \sin \theta)), and letting N = \left( \frac{4}{5}, \frac{3}{5} \right) be the unit normal we have the polar equation

    \begin{align*} && \lVert X \rVert &= e | X \cdot N - d | \\[9pt] \implies && r &= \left| \frac{4}{5} r \cos \theta + \frac{3}{5} r \sin \theta - 5\right| \\[9pt] \implies && r &= 5 - \frac{4}{5} r \cos \theta - \frac{3}{5} r \sin \theta \\[9pt] \implies && 5r &= 25 - 4 r \cos \theta - 3 r \sin \theta \\[9pt] \implies && r &= \frac{25}{5 + 4 \cos \theta + 3 \sin \theta}. \end{align*}

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