### Polar Coordinates - **Conic Sections in Polar Form (Focus as Pole):** - Polar equations for parabola, ellipse, and hyperbola. - **Chord, Tangent, and Normal:** - Polar equations of the chord joining two points. - Tangent and normal lines. - Polar (chord of contact). - Pair of tangents. - **Related Concepts:** - Asymptotes. - Director circle. - Auxiliary circle of a conic. ### Generating Lines - **Ruled Surfaces:** - Definition and properties. - **Hyperboloid of One Sheet:** - Generating lines and their properties. - Generators through a point on the principal elliptic section. - $( \theta, \phi )$ point on hyperboloid of one sheet. - Equations of generators at $( \theta, \phi )$ point. - **Hyperbolic Paraboloid:** - Generating lines and their properties. ### Central Conicoids & Paraboloids - **Standard Equations:** - Standard equations for central conicoids and paraboloids. - **Tangent Lines and Planes:** - Tangent lines and planes. - **Polar Planes and Lines:** - Polar planes and polar lines. - **Enveloping Cones and Cylinders:** - Formation and properties. - **Section with Given Centre:** - Analysis of sections with a specified center. - **Diametral Planes:** - Diametral planes of central conicoids and paraboloids. - **Locus of Intersection of Tangent Planes:** - Locus of the point of intersection of three mutually perpendicular tangent planes to central conicoids and paraboloids. - **Normals:** - Normal drawn from a point to a central conicoid and a paraboloid. - Cone passing through the normals drawn from a point to a central conicoid and paraboloid. - **Conjugate Diameters of an Ellipsoid:** - Definition and properties.
