ASSIGNMENT BOOKLET. Bachelor s Degree Programme. Analytical Geometry

Transcription

1 ASSIGNMENT BOOKLET MTE-05 Bachelor s Degree Programme Analytical Geometry (Valid from st January, 0 to st December, 0) School of Sciences Indira Gandhi National Open University Maidan Garhi New Delhi-0068 (For January 0 cycle)

2 Dear Student, Please read the section on assignments in the Programme Guide for Elective courses that we sent you after your enrolment. A weightage of 0 per cent, as you are aware, has been earmarked for continuous evaluation, which would consist of one tutor-marked assignment for this course. The assignment is in this booklet. Instructions for Formatting Your Assignments Before attempting the assignment please read the following instructions carefully. ) On top of the first page of your answer sheet, please write the details eactly in the following format: ROLL NO: NAME: ADDRESS: COURSE CODE:. COURSE TITLE:. ASSIGNMENT NO.. STUDY CENTRE:.... DATE:.... PLEASE FOLLOW THE ABOVE FORMAT STRICTLY TO FACILITATE EVALUATION AND TO AVOID DELAY. ) Use only foolscap size writing paper (but not of very thin variety) for writing your answers. ) Leave cm margin on the left, top and bottom of your answer sheet. ) Your answers should be precise. 5) While solving problems, clearly indicate which part of which question is being solved. 6) This assignment is to be submitted to the Study Centre as per the schedule made by the study centre. Answer sheets received after the due date shall not be accepted. We strongly suggest that you retain a copy of your answer sheets. 7) This assignment is valid only upto December, 0. If you have failed in this assignment or fail to submit it by December, 0, then you need to get the assignment for the year 0 and submit it as per the instructions given in the programme guide. 8) You cannot fill the eam form for this course till you have submitted this assignment. So solve it and submit it to your study centre at the earliest. We wish you good luck.

3 Assignment (MTE-05) (January 0 December 0) Course Code: MTE-05 Assignment Code: MTE-05/TMA/0 Maimum Marks: 00. Check whether the following statements are true or false. Give valid reasons to support your answer. (0) i) y z represents a line. ii) iii) A section of a paraboloid may have more than one centers. Intersection of a cone and a plane cannot be a single point. iv) Polar coordinates of ( 0, ) are (, / ). v) The equation vi) 9 y Any two conics intersect in four real points. represents a hyperbola if 0. y z vii) The equation f (, y, z) c 0, where f is a linear homogeneous function in, y and z and c is a constant, does not represent a sphere. viii) i) A cone is a conicoid but a conicoid may not be a cone. Intersection of a plane and a non-central conicoid is always a non central conic.. ) The section of the conicoid y z by the plane z 8 is a circle.. a) Obtain the equation of a line passing through (, ) and perpendicular to the line 7y 8 0. Epress the obtained equation into the intercept form. () b) Show that the line segments joining the points (, ), (, ) and (, ), (0, - ) bisect each other. Also show that the quadrilateral formed by taking these points as the vertices is a parallelogram. (5) c) Find the length of the major and minor aes, the eccentricity, coordinates of the vertices and the foci of 9y. Also draw a rough sketch of it. () d) What will be the equation of a hyperbola with vertices ( ±, 0) and transverse and conjugate aes having the ratio :? Derive it. (). a) What type of conic is represented by the equation 8 y y 6y 0? Discuss. () b) Check whether the conic 7 6y y 0 6y 0 is central or not. Find its centre in case if it is central. Also trace it. () c) Check whether the point (,, ) lie on the plane passing through the points (,, ), (,, ) and (, 6, 8). ()

4 . a) Check whether the line cos ysin p is tangent to the conic y p or not. Verify your answer geometrically. () b) Give an eample of two conics which intersect each other at a single point. () c) Let A (, y, z), B(, y, z ) and C (, y, z) be the vertices of a triangle. Find the angle between the line BC and the line passing through the mid-point of AB and AC. What do you conclude from this. () y d) Find, if any, point(s) of intersection of the line z and the plane 9 5y z. () 5. a) Does any sphere pass through the points (,, ), (,, ), (, 5, 0) and ( 0,, )? If yes, find its equation. (5) b) For which value of a, the line a y z is tangent to the sphere y z 7y z 0? Find the point of tangency also. () c) Find the equation of a cone whose verte is at the origin and base curve is a y bz, y z 0. When this cone will be a circular cone? () d) Find the reciprocal cone of 6 z y 5yz 0. () 6. a) Find the equation of the cylinder which passes through the circle ( ) y z 8, y z. () b) Find the centre of the conicoid y y 6z 7 8 y 8 0, if it eists. Transform this conicoid by shifting the origin to its centre. (5) c) What is the projection of the line segment joining (,,5) and (,, ) on the line ( ) y z. () d) Sketch the surface defined by the equation 9 5y 55( z ). Geometrically describe the sections of this surface by the plane i) z 0 ii) z (5) 7. a) Transform the plane y z under the transformations given in the following table y z y z b) Identify the type of the conicoid y z. () c) What object is formed by the intersection of the curves 5y z and z 5( y )? Justify your answer. () 7 6 ()

5 d) Which of the following are paraboloids and which are parabolic cylinders? Further if any one is a paraboloid, classify it as elliptic paraboloid or hyperbolic paraboloid. i) ( y z ) ( y z) 0 ii) y 9z 5 0 iii) 7 z y z 0. (++) 5