Focus-directrix conics in polar form

Connect focus-directrix geometry with polar equations of conics. Normalize denominators to identify eccentricity, locate a compatible directrix, and find vertices relative to a focus at the pole rather than the center. Single-select and multiselect tasks include derivation, Cartesian verification and explicit nonnegative-radius branch restrictions. Detailed explanations highlight sign conventions, excluded denominator zeros and the special status of the circular limit in the finite-directrix definition.