TOPIC CONE DEFINITION OF CONE HOMOGENEOUS EQUATION OF

TOPIC : CONE

DEFINITION OF CONE

HOMOGENEOUS EQUATION OF CONE

EQUATION OF CONE WITH VERTEX AT ORIGIN

EQUATION OF A CONE

EXAMPLES RELATED TO CONE

EQUATION OF CONE WITH A GIVEN VERTEX AND BASE A GIVEN CONIC

RIGHT CIRCULAR CONE

EQUATION OF RIGHT CIRCULAR CONE (STANDARD FORM)

EXAMPLES

ENVELOPING CONE The locus of tangent drawn from a fixed point to a sphere (or conicoid) is a cone , called enveloping cone or tangent from the point to the sphere.

EQUATION OF ENVELOPING CONE

ENVELOPIG CONE IN GENERAL FORM

EXAMPLES RELATED TO ENVELOPING CONE

INTERSECTION OF A STRAIGHT LINE AND A CONE

EQUATION OF THE TANGENT PLANE

ASSIGNMENT

n n Find the equation to the cone with vertex at the origin and which passes through the curves x 2+y 2=4, z=2 Find the equation to the cone with vertex at the origin and which passes through the intersection of the surfaces 4 x 2+y 2+z 2+2 x+4 z=8 4 x 2+y 2+z 2+x+2 z=4

n n n Find the equation of the cone whose vertex is( , , ) and whose base is y 2=4 ax, z=0 Find the equation of the cone whose vertex is( , , ) and whose base is ax 2+by 2=1, z=0 Find the equation of the cone whose vertex is the point (1, -1, 2) and whose guiding curve is 3 x 2 -y 2=1, z=0

n n n Find the equation to the right circular cone whose vertex is p(2, -3, 5), axis pq which makes equal angles with the co-ordinates axis and semi-vertical angle is 30. Find the enveloping cone of the sphere x 2+y 2+z 2+2 x-2 y=2 with its vertex at (1, 1, 1) Prove that 4 x 2 -y 2+2 z 2+2 xy-3 yz+12 x-11 y+6 z+4=0 represent a cone whose vertex is (-1, -2, -3).

TEST n NOTE: DO ANY TWO

n n n Find the equation to the cone, whose vertex is at the origin and which passes through the curves x 2/4+y 2/9+z 2/1=1 x+y+z=1 Find the equation of the conewith vertex (1, 1, 1) and passes through the curve of intersection of x+y+z=1 and x 2+y 2+z 2=1 Find the equation to the right circular cone, whose vertex is p(2, -3, 5), axis pq which makes equal angles with the axes and which passes through a(1, -2, 3).