Course Outlines

·            Cone with vertex at origin

·            Condition for the general equation of second degree to represent a cone

·            Coordinates of the vertex of a cone

·            Equation of a cone with a given vertex and given conic as base

·            Angle between the lines in which a plane cuts a curve

·            Condition that a curve has three mutually perpendicular generators  

·            Tangent lines and tangent plane at  a point

·            Condition for tangency

·            Equations of reciprocal cone, enveloping cone and right circular cone, enveloping cylinder and  right circular cylinder.

Cone

        A cone is a surface generated by a straight line which passes through a fixed point, and satisfies one or more condition; for instant, it may intersect a given curve or may touch a given surface. The fixed point is called the vertex and the given curve the guiding curve of the cone is called of the cone.

        An individual straight line on the surface of a cone is called its generator. Thus, a cone is essentially a set of lines called generators through a given point. Also we may say that a cone is a set of points on its generators. Whereas we can have cones with equations of any degree whatsoever depending upon the condition to be satisfied by its generators, we shall in here be concerned only with quadratic cones, i.e. cones will second degree equations.

        It will be seen that the degree of the equation of a cone whose generators intersect a given conic or touch a given sphere is of the second degree.