This document discusses different types of conicoids and their properties. It defines a conicoid as a surface whose cross sections are conic sections. There are four types of central conicoids based on the signs of coefficients in their equation: ellipsoid, hyperboloid of one sheet, hyperboloid of two sheets, and virtual quadric. Properties discussed include plane sections, tangency, enveloping cones/cylinders, polar planes, and paraboloids. Conjugate points, lines, and diametral planes are also defined. The document provides mathematical conditions and definitions for many geometric properties and concepts relating to conicoids.

1. WELCOME

2. Chapter 1
conicoid

3. Conicoid
A surface whose equation is of
second degree is called a conicoid. i.e.,
A surface of which every plane section is
a conic.

4. central conicoid
A conicoid all of whose chords
through the origin are bisected at the
origin is called a central conicoid and is
represented by the equation of the form
ax2+by2+cz2=1

5. TYPES
1. If all the three signs are positive, then
the surface is called ellipsoid;
2. If one of the signs is negative, the
surface is called the hyperboloid of one
sheet;
3. If two of the signs are negative, the
surface is called the hyperboloid of two
sheets;
4. If all the three signs are negative, the
surface is called the virtual quadric.

6. Nature of plane sections of central conicoid
The plane section of the central conicoid is an ellipse if
c2l2m2<(an2+cl2)(bn2+cm2) ;
The projection is a parabola if c2l2m2=(an2+cl2)(bn2+cm2) ;
The projection is a hyperbola if c2l2m2>(an2+cl2)(bn2+cm2) .

7. Tangency:
A line which meets a conicoid in two
coincident points is called the tangent
line to the conicoid.
The locus of the tangent lines to the
conicoid at a point on it is called the
tangent plane at that point.

8. Condition for the plane lx+my+nz = p
to be a tangent plane to the
conicoidp = ±
𝑙2
𝑎
+
𝑚2
𝑏
+
𝑛2
𝑐
.
Condition of a plane lx+my+nz=p to
be tangent plane to the ellipsoid
p=± 𝑎2 𝑙2 + 𝑏2 𝑚2 + 𝑐2 𝑛2

9. Enveloping cone
The locus of the tangent lines drawn
from a given point to a conicoid is a
cone and is called a tangent cone or
enveloping cone.The given point is
called the vertex of the cone.

10. Enveloping cylinder
The locus of tangent line drawn to a
conicoid and parallel to a given line is a
cylinder and is called enveloping
cylinder of the conicoid.

11. Polar Plane
The polar plane of an external point P
with respect to a quadric is the plane
which contains the points of contact of
all the tangent lines drawn from P to the
quadric.

12. Conjugate points and conjugate
planes
If the polar plane of a point P passes
through the point Q , then the polar
plane of Q passes through the point P. Two
such points are known as conjugate
points and two such planes are called
conjugate planes .

13. Polar lines or Conjugate lines
Two lines such that the polar plane of any
point lying on one line passes through the
other line are called conjugate lines or
polar lines. Therefore the polar line of any
given line is the line of intersection of the
polar planes of any two points on the given
line.

14. Paraboloid
A solid generated by the rotation of the
parabola about its axis of symmetry is
called paraboloid. In other words,
paraboloid is a solid having more than
two non parallel parabolic cross
sections.

15. TYPES
(1).Both the terms are of the same
sign, the surface of the ellipse is called
elliptic paraboloid.
(2).The terms are of the opposite
signs, the surface is called hyperbolic
paraboloid.

16. Diametralplanes
A line through the centre of the conicoid
is called a diameter of the conicoid.
A plane through the centre of the conicoid
is called a diametral plane of the conicoid.
The three diametral planes, which are
such that each is the diametral plane of the
line of intersection of the other two are
called conjugate diametral planes.

17. Submitted by,
k.mythili,
1st year B.ed,
Dr.Sivanthi aditanar
college of education

18. THANK YOU