Cone Def 1. Cone : A cone is a surface generated by a straight line which passing through a fixed point and satisfies one more condition. (for instance , it may intersect a given curve or touch a given surface). The fixed point is called the vertex and the moving straight line is called generator and the given curve is called the guiding curve of the cone. Remarks.1 There is no loss of generality in taking the guiding curve as a plane curve because any arbitrary plane section of the surface can be taken as a guiding curve.

(2) Through a cone lies on both sides of the vertex, we for the sake of convenience show only one side of it in the figure. Def 2 Homogenous equation: An equation f(x,y,z)=0 in three unknowns x , y , z is called homogenous if (tx ,ty, tz)=0 for all real numbers t. Example: The equation is homogenous. Example : Show that is not homogenous.

Equation of a cone whose vertex is the origin. Theorem: The equation of a cone , whose vertex is the origin is homogenous in x,y,z and conversaly any homogenous x ,y ,z represents a cone whose vertex is the origin. Equation of a cone with a conic as guiding curve To find the equation of the cone whose vertex is the point and whose generator intersect the conic. Elliptic cone. Def 4. An elliptic cone is a quadric surface which is generated by a straight line which passes through a fixed point and which intersect an ellipse.

(b) Equation of an elliptic cone: To find the equation of the ellpitic cone whose vertex is the origin and which intersects the ellipse. Example 3 Find the equation to cone whose vertex is the origin and which passes through the curve of intersection of the plane. and the surface Example 4 Planes through OX,OY include an angle Show that the line of intersection lies on the cone. Example 5. Prove that the line x=pz+q , y= rz+s intersects the conic z=0,

Theorem If is a generator of the cone represented by homogenous equation f(x,y,z)=0, then f(l, m, n) =0 Or The director – cosine of a generator of a cone, whose equation is homogenous, satisfy the equation of the cone. Example: Show that the line where Is a generator of the cone Def. Quadratic cone A cone whose equation is of the second degree in x, y, z is called a quadric cone.

Condition for a general equation of second degree to represent a cone. Theorem: To find the condition that the equation may represent a cone. If the condition is satisfied, then find the co-ordinate of the vertex. Example: prove that the equation

General equation of a quadric cone through the axes: To show that the general equation of the cone of second degree, which passes through the axes, is fyz + gzx + hxy=0 Enveloping cone:- The locus of the tangents from a given point to a sphere (or a conicoid ) is a cone called the enveloping cone from the point to the sphere (or conicoid). It is also called the enveloping cone with the given point as the vertex. Theorem: To find the equation of the enveloping cone of the sphere with the vertex at P

Find the enveloping cone of the sphere with its vertex at (1,1,1) Intersection of a straight line and a cone. To find the points where the line are meets the cone Tangent plane: To find the equation of the tangent plane at the point P( ) of the cone Example: Show that the line is the line intersection of tangent plane to the cone

along the line which it is cut by the plane. Condition that a cone may have three mutually perpendicular generators. may have three mutually perpendicular generators. or Show that the cone has three mutually perpendicular generators iff a+b+c=0 Example: Show that the three mutually perpendicular tangent lines can be drawn to the sphere From any point on the sphere

Condition of tangency of a plane and a cone To find the condition that the plane lx + my + nz =0 Should touch the cone Reciprocal cone The locus of the normals through the vertex of a cone to the tangent planes is another cone which is called the reciprocal cone. Theorem: to find the equation of the reciprocal cone to the cone Def: Reciprocal cones : Two cones , which are such that each is the locus of the normals through the vertex to the tangent planes to the other, are called reciprocal cones.

Condition that a cone may have three mutually perpendicular tangent planes. To prove that the condition , that the cone May have three mutually perpendicular tangent planes, is A + B +C =0 Where Example: Show that the general equation of the cone which touches the three co-ordinate planes is where f, g, h are parameters.

Prove that the semi-vertical angle of a right circular cone admitting sets of three mutually perpendicular generators is Right circular cone with the vertex at the origin, a given axis and a given semi-vertical angle Show that the equation of the right circular cone whose vertex is the origin, axis the line (l, m, n being direction - cosine) and semi-vertical angle is Right circular cone with a given vertex , a given axis and a given semi- vertical angle. Find the equation of the right circular cone whose3 vertex is semi-vertical angle and axis has the direction cosine<l, m, n>

Example : Find the equation of the right circular cone whose vertex is P(2,-3,5), axis makes equal angles with the co-ordinate axes and semi-vertical angle Example: Find the equation of the right circular cones which contain three co-ordinate axes as generators.