Syllabus B. Ed. Third Year
Course
Title: Analytical
Geometry Full
marks: 100
Course
number: Math. Ed. 431 Major Pass marks: 35
Nature
of course: Theory + Practical Period
per weeks: 6
Level:
Bachelor Degree Total
Period:150
1. Course
Description
This course Analytic Geometry deals with the properties
of geometric figures using coordinate system. It is concerned with two or three
dimensions in which students will be able to generalize the nature and
properties of geometric shapes using algebraic properties. Analytic geometry is
widely used in various branches of science. This geometry also creates the
foundation of most modern fields of geometry including algebraic differential, discrete
and computational geometry. It is concerned with defining and refreshing geometrical
shapes in a numerical way and extracting numerical information from shape’s
numerical definitions and representations.
2. General objectives
The
general objectives of this course are as follows:
·
To familiarize
students with different co-ordinate systems in analytical geometries of two and
three dimensions.
·
To make the
students able to understand different conic sections and describe their natures.
·
To acquaint
students in describing analytically the structure of space, special relation
with lines, planes and relations between them in 3-space.
·
To make students
able to generalize the general equation of second degree and conditions to
represent conics and conicoids with their properties.
·
To make a deep
understanding of plane sections and generating lines of conicoids.
3 .Specific
Objectives and Contents
|
Specific Objectives |
Contents |
|
|
·
Describe the
transformation of coordinates of axes ·
Derive
transformation of coordinates of axes through translation and rotation ·
Define invariants
in orthogonal transformations |
Unit-I Transformation of Co-ordinates (6) 1.1 Translation
of axes 1.2 Rotation
of axes 1.3 Combination of
translation and rotation of axes, 1.4Invariants in orthogonal
transformation. |
|
|
·
Discuss the
nature of curves obtained by conic sections namely parabola, ellipse,
hyperbola and interpretation as a locus of a point ·
Determine the
equation of parabola, ellipse and hyperbola and discuss their nature. ·
Derive
equations of tangents and normal to parabola, ellipse and hyperbola and
discuss their properties ·
Define pole
and polar on parabola, ellipse and hyperbola |
Unit-II Conic Sections (30) 2.1 Parabola: 2.1.1 Introduction of conic sections 2.1.2 Review of
equations of tangent and normal 2.1.3 Equations of
pair of tangents, 2.1.4 Director circle 2.1.5 Chord of
contact 2.1.6 Pole and polar 2.1.7 Properties of
pole and polar. 2.2 Ellipse 2.2.1 Equation of
ellipse 2.2.2 Auxiliary
circle and eccentric angles 2. 2.3 Position of a point 2. 2.4 Tangent and
normal 2. 2.5 Pair of tangents from an external point 2. 2.6 Director circle 2. 2.7 Chord of
contact, 2. 2.8 Pole and polar
and their properties 2. 2.9 Chord with a given
middle point 2. 2.10 Diameter, Conjugate
diameter 2. 2.11 Equiconjugate
diameters and its properties 2.3 Hyperbola 2.3.1 Equation of hyperbola 2.3.2Parametric
coordinates and conditions of tangency 2.3.3 Equation of
tangent and normal 2.3.4 Chord or
contact 2.3.5 Pair of
tangents 2.3.6 Auxiliary
circle and director circle 2.3.7Conjugate points
and lines 2.3.8Equation of
chord 2.3.9 Rectangular
hyperbola 2.3.10 Asymptotes and
its equations 2.3.11 Equation of diameter
and its properties 2.3.12Conjugate diameters
and its properties. |
|
|
·
Represent
conics in polar coordinates. ·
Derive equations
of conics in polar coordinates. ·
Trace conics
in polar form. ·
Determine
equations of chord, tangent and normal. |
Unit-III Polar Equation of a
Conic (12) 3.1Polar co-ordinate
system 3.2 Polar
equation of conics 3.3 Equation
of directrix 3.4Tracing conics in
polar form 3.5Equations of
chord, tangent and normal 3.6 Point of
intersection of two tangents 3.7 Equation of pair of tangents 3.8 Equation
of chord of contact. |
|
|
·
Derive the
representation of general equation of second degree as a conic section under certain
conditions. ·
Discuss
different properties of conic section of the second degree equation. ·
Determine
equations of asymptotes, tangents, and normal, director circle, and chord of
contact. ·
Discuss pole,
poler and chord of the general conic. |
Unit-IV Conic Sections Represented by
General Equation of Second Degree (18) 4.1General equation
of second degree and conics represented by this equation 4.2 Nature and centre
of conic 4.3 Reduction of
centre of conic to standard form 4.4 Equation
of asymptotes 4.5 Equation of
tangent and normal 4.6Conditions of tangency 4.7 Pair of tangents
from an external point 4.8 Equation of director
circle and chord of contact 4.9 Pole and polar and their properties 4.10 Chord of the
general conic with given middle point 4.11 Diameter
of the conic 4.12 Conjugate
diameters 4.13 Intersection
of conics 4.14 Equation of
conic through the intersection of two conics. |
|
|
·
Review the
rectangular Cartesian co-ordinates in 3-dimension, change of origin, section
formula, direction cosines, direction ratios, projection and angle between
two lines ·
Relate
Cartesian co-ordinates, spherical co-ordinates and cylindrical co-ordinates
of a point. ·
Define plane
in 3D and establish linear equation representing a plane. ·
Find equation
of plane in intercept form, normal form and reduce general equation of plane
in normal form. ·
Determine
plane through three points, plane through the intersection of two planes. ·
Determine
angle between two planes and plane bisecting the angle between two planes. ·
Establish
condition for homogeneous equation to represent a pair of planes |
Unit-V Plane
(15) 5.1 Review the three
dimensional Cartesian co-ordinates 5.2 Cylindrical and
spherical co-ordinates of a point 5.3
General equation of first degree 5.4
Linear equation of a plane 5.5
Angle between two planes 5.6
Angle between a line and a plane 5.7
Plane through three points 5.8 Plane through the
intersection of two planes 5.9 Length of
perpendicular from a point to a plane 5.10 Bisectors of
angles between two planes 5.11
Pair of planes 5.12 Conditions for
homogeneous second degree equation to represent a pair of planes 5.13Angle between two
planes represented by a second degree homogeneous equation. |
|
|
·
Derive the
equation of straight line in symmetrical form and equation of straight line
joining two points. ·
Transform
general equation to symmetrical form. ·
Find angle
between a line and a plane. ·
Derive the
condition for a line to lie in a plane. ·
Derive the
condition for co- planarity of lines. ·
Find the
shortest distance between two lines. |
Unit-VI
Straight Lines (17) 6.1 Equation of a straight
line in symmetrical form 6.2 Perpendicular
distance of a line from a point 6.3
Two forms of the equation of a line 6.4
Angle between a line and a plane 6.5Condition for a line to lie in a plane 6.6
Plane containing a line coplanar lines,
6.7 Shortest distance between two lines. |
|
|
·
Determine equation of a sphere in
different conditions. ·
Determine the intersection of two
spheres. ·
Discuss the intersection of a
sphere and a time. ·
Find the equation of a tangent
plane and determine the condition of tangency. |
Unit-VII
Sphere (10) 7.1 Equation of a sphere 7.2 General equation of a sphere 7.3 Equation of a sphere
through four points 7.4 Plane section of a sphere 7.5 Equation of a sphere with a given diameter 7.6 Intersection of a two spheres 7.7 Spheres through the given circle 7.8 Intersection of a sphere and a
line 7.9 Equation of tangent plane 7.10 Condition of tangency |
|
|
·
Find equation of a cone and
cylinder ·
Determine the condition of
general equal of second degree to represent a cone. ·
Find equation of a cone with a
generic conic as a base, ·
Find angles between two lines in
which a plane cuts a cone, ·
Find equation of tangent lines, planes and condition of
tangency, ·
Find equation of a reciprocal
cone, enveloping cone and right circular cone and cylinder. |
Unit-VIII
Cone and Cylinder (10 ) 8.1 Cone with vertex at origin 8.2 Condition for the general equation of second degree to represent
a cone 8.3 Coordinates
of the vertex of a cone 8.4 Equation
of a cone with a given vertex and given conic as base 8.5 Angle between the lines in which a plane cuts
a curve 8.6 Condition that a curve has three mutually
perpendicular generators 8.7 Tangent lines and
tangent plane at a point 8.8 Condition for tangency 8.9 Equations of reciprocal cone, enveloping cone and right
circular cone, enveloping cylinder
and right circular cylinder. |
|
|
·
Write equation and identify the
shapes of ellipsoid, hyperboloid of one-sheet and two-sheets. ·
Find equation of a line with a
conicoid. ·
Find equations of tangent planes. ·
Find equation of normal from a
given point, cubic curves through the feet of the normal and cone through six
normals. ·
Derive equation of polar plane of a point
and find the pole of a given plane, ·
Find equation of enveloping cone
and cylinder ·
Find diametrical plane, principal
plane and conjugate diameters of the ellipsoid. |
Unit-IX
Central Conicoid (12) 9.1Equations
and shapes of ellipsoid and
hyperboloid 9.2 Intersection of a line with a conicoid 9.3 Equation of a tangent plane 9.4 Condition of tangency 9.5 Equation of normal 9.6 Cubic curves through the feet of the normals and
cone
through six normals, 9.7 Director sphere 9.8 The plane of contact 9.9 Polar plane of a point 9.10 Pole of a given line 9.11 Properties
of polar planes and polar lines 9.12 Locus
of chords bisected at a given point 9.13 Locus of middle points of a system of
parallel chord 9.14 Enveloping cone and enveloping cylinder 9.15 Diametral plane and principal plane 9.16 Conjugate
diameter and conjugate diametral planes of ellipsoid 9.17 Properties
of conjugate semi-diameters. |
|
|
·
Discuss the plane sections of
conicoids ·
Determine the nature ,the lengths
,and the direction ratios of the axes of a
plane of a given conicoid ·
Discuss generating lines and
condition for a line to be a generator of conicoid ·
Deal with properties of
generating lines of hyperboloid of one sheet ·
Discuss the generating lines of a
hyperbolic paraboloid and its properties |
Unit-X Plane sections and Generating Lines of
Conicoids (20) 10.1 Nature of the plane sections of a central conocoid 10.2 Axes of acentral plane section 10.3Areas of
plane sections 10.4 Condition
for the section to be a rectangular hyperbola 10.5 Axes of non-central plane sections 10.6 Parallel
plane sections 10.7 Circular sections 10.8 Umbilics 10.9 Axes of plane sections of paraboloids 10.10 Circular sections of paraboloids 10.11Generating lines of a hyperboloid one sheet 10.12Condition for a line to be a generator of the
conicoid 10.13 Properties of generating lines of
hyperboloid of one sheet 10.14 Projections of the generators of a hyperboloid on any principal plane 10.15 Perpendicular generators 10.16 Properties
of generating lines of hyperbolic
paraboloid. |
|
4. Instructional Techniques
The
nature of this course being theoretical, teacher-centred instructional
techniques will be dominant in teaching-learning process. The teacher will
adopt the following techniques.
4.1 General instructional
techniques
·
Lecture with illustration
·
Discussion
·
Demonstration
4.2
Specific instructional techniques
·
Inquiry and question-answer (for all
units)
·
Assignment and presentation (for all
units)
·
Individual and group work presentation
5. Evaluation
Students
will be evaluated on the basis of written test in between and at the end of the
academic session, the classroom participation, presentation of the assignment (reports)
and other activities. The scores obtained will be used only for feedback
purposes. The office of the controller of examination will conduct annual examination
at the end of the academic session to evaluate student's performance. The
types, number and marks of the subjective and objective questions will be as
follows:-
|
Types
of questions |
Total
questions to be asked |
No.
of questions to be answered and allotted |
Total
marks |
|
Group A: Multiple choice items |
20 questions |
20 |
20 |
|
Group B: Short questions |
8 with 3 alternative questions |
8 |
56 |
|
Group C: Long questions |
2 with 1 alternative question |
2 |
24 |
6.
Recommended and Reference Books
6.1
Recommended Books
Koirala
S.P., Pandey U.N. & Pahari N.P. (2009), Analytic
geometry Kathmandu; Vidyarthi Prakashan (P) Ltd. (Third revised ed. 2016).
(For all units)
Joshi
M.R. (1997); Analytic geometry, Kathmandu;
Sukunda Pustak Bhandar
Loney
S.L. (1984): The elements of coordinate
geometry; New Delhi: S. Chand and company Pvt. Ltd.
6.2 Reference
Books
Chatterjee, D. Analytical solid geometry,New Delhi
:Prentice Hall of India Private limited.
Narayan
S. and Mittal P. K. (2001), Analytical
Solid geometry, New Delhi: S. Chand
and Company Pvt. Ltd.
Pandit,
R.P. and Pathak, B. R., (2069) Fundamentals
of geometry Kathmandu: Indira Pandit.
Prasad
Lalji, (1990). Analytical Solid geometry,
Panta: Paramount Publication
Sthapit,
Y. R. & Bajracharya, B.C. (1992). A
textbook of three dimensional geometry. Kathmandu: Sukunda Pustak Bhandar.
Thomas,
G. B. & Finney R. L. (2004), Calculus
and analytic geometry New Delhi: Pearson publication.
Mittal
P.K.(2007).,Analytical geometry. Delhi:
Vrinda publications (P) LTD
2 Comments
Great guru
ReplyDeletesubject matter of analytical geometry will be published soon.
Delete