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
|
Contents |
|
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. |
|
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 |
|
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. |
|
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. |
|
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. |
|
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. |
|
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 |
|
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. |
|
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. |
|
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
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