Cone Def 1 Cone A cone is a

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 of tangent plane to the cone is the line intersection

along the line which it is cut by the plane. • Condition that a cone 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 whose 3 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.