Transcript Cone - GCG-42

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
ax2  by 2  cz 2  2 fyz  2gzx  2hxy  0 is homogenous.
Example : Show that
2 x2  3 y 2  x  4 y  z  0 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.
2
2
2
ax  by  cz  2 fyz  2gzx  2hxy  0, Z  0
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. lx  my  nz  1
and the surface ax2  by2  cz2  1
Example 4 Planes through OX,OY include an angle 
Show that the line of intersection lies on the cone.
z 2 ( x2  y 2  z 2 )  x2 y 2 tan2 
Example 5. Prove that the line x=pz+q , y= rz+s
intersects the conic z=0, ax2  by2  1ifaq2  bs2  1
x y z
  t
l m n
• 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 xl  my  nz where l 2  2m2  3n2  0
Is a generator of the cone x  2 y  3z  0
Def. Quadratic cone A cone whose equation is of the
second degree in x, y, z is called a quadric cone.
2
2
2
• Condition for a general equation of second degree to
represent a cone.
• Theorem: To find the condition that the equation
 yxh2  xzg 2  zyf 2  2 zc  2 yb  2 xa  ) z ,y ,x ( F
0  d  zw2  yv2  xu2
may represent a cone. If the condition is satisfied,
then find the co-ordinate of the vertex.
Example: prove that the equation
x2  2 y 2  3z 2  4 xy  5 yz  6zx  19 y  2z  20 
• 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 x2  y2  z2  a2 with the vertex at P ( x1, y1, z1)
• Find the enveloping cone of the sphere
x2  y 2  z 2  2 fyz  2 x  2 y  2 with its vertex at (1,1,1)
Intersection of a straight line and a cone. To find the points
where the line are
x  x1 y  y1 z  z1


r
meets the cone
l
m
n
ax2  by 2  cz 2  2 fyz  2gzx  2hxy  0
Tangent plane:
To find the equation of the tangent plane at the
point P( x1, y1, z1 ) of the cone
ax2  by 2  cz 2  2 fyz  2gzx  2hxy  0
Example: Show that the line xl  my  nz is the line intersection
of tangent plane to the cone
ax2  by 2  cz2  2 fyz  2gzx  2hxy  0
along the line which it is cut by the plane.
• Condition that a cone may have three mutually
perpendicular generators.
ax2  by 2  cz2  2 fyz  2gzx  2hxy  0 may have three mutually
perpendicular generators.
or
2
2
2
ax

by

cz
 2 fyz  2gzx  2hxy  0 has three
Show that the cone
mutually perpendicular generators iff a+b+c=0
Example: Show that the three mutually perpendicular
tangent lines can be drawn to the sphere x2  y2  z2  r2
2
2
2
2
2(
x

y

z
)

3
r
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
ax2  by 2  cz 2  2 fyz  2gzx  2hxy  0
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 ax2  by2  cz2  2 fyz  2gzx  2hxy  0
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
ax2  by 2  cz 2  2 fyz  2gzx  2hxy  0
May have three mutually perpendicular tangent planes,
is A + B +C =0
A  bc  f 2 , B  ca  g 2 , C  ab  h 2or
Where
2
2
2
bc  ca  ab  f  g  h
Example: Show that the general equation of the cone
which touches the three co-ordinate planes is
fx  gy  hz  0 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 tan1 2
• 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 xl  my  nz
(l, m, n being direction - cosine) and semi-vertical angle
 is (lx  my  nz)2  ( x2  y2  z2 )cos2 x
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 A( ,  ,  ) 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
30 angle
with the co-ordinate axes and semi-vertical
• Example: Find the equation of the right circular
cones which contain three co-ordinate axes as
generators.
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