Exercises 29-36 give the eccentricities of conic sections...

Exercises $29-36$ give the eccentricities of conic sections with one
focus at the origin along with the directrix corresponding to that focus.
Find a polar equation for each conic section.
$$
e=5, \quad y=-6
$$

Key Concepts

Polar Conic Section Equation

In polar coordinates, conic sections are represented by an equation of the form r = ed/(1 + e cos ?) or r = ed/(1 + e sin ?), depending on the orientation of the directrix. This formulation places one focus at the origin and expresses the distance r as a function of the polar angle ?, with the constants e (eccentricity) and d (the distance from the focus to the directrix) determining the shape and position of the conic.

Eccentricity

Eccentricity is a fundamental parameter that describes the shape of a conic section. It is defined as the constant ratio of the distance from any point on the conic to the focus, over its distance to the corresponding directrix. Values of eccentricity classify the conic: less than 1 for ellipses, equal to 1 for parabolas, and greater than 1 for hyperbolas.

Focus-Directrix Relationship

The focus-directrix property is the defining characteristic of conic sections, stating that for any point on the curve, the ratio of its distance to the focus and its perpendicular distance to the directrix remains constant (and equal to the eccentricity). This relationship forms the basis for deriving the polar equations of conics by linking geometric distances to algebraic expressions in r and ?.

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