CO ORDINATE GEOMETRY

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MATHS SERIES,CO ORDINATE,FIREWALL MEDIA,An Imprint of Laxmi Publications Pvt Ltd. BANGALORE l CHENNAI l COCHIN l GUWAHATI l HYDERABAD. JALANDHAR l KOLKATA l LUCKNOW l MUMBAI l RANCHI,NEW DELHI l BOSTON USA. 4 D L gcorg cogtit cog IInd 16 10 10, Copyright 2013 by Laxmi Publications Pvt Ltd All rights reserved No part. of this publication may be reproduced stored in a retrieval system or. transmitted in any form or by any means electronic mechanical photocopying. recording or otherwise without the prior written permission of the publisher. Published by,FIREWALL MEDIA,An Imprint of Laxmi Publications Pvt Ltd. 113 Golden House Daryaganj,New Delhi 110002,Phone 011 43 53 25 00.
Fax 011 43 53 25 28,www laxmipublications com,info laxmipublications com. Price 495 00 Only First Edition 2008 Reprint 2009 2010 2011. Second Edition 2013, Bangalore 080 26 75 69 30 Jalandhar 0181 222 12 72. Chennai 044 24 34 47 26 Kolkata 033 22 27 43 84, Cochin 0484 237 70 04 405 13 03 Lucknow 0522 220 99 16. Guwahati 0361 251 36 69 251 38 81 Mumbai 022 24 91 54 15 24 92 78 69. Hyderabad 040 24 65 23 33 Ranchi 0651 220 44 64,FCO 3076 495 G CO ORDINATE GEO BAL C 2023 10 08. Typeset at Goswami Associates Delhi Printed at Ajit Printers Delhi. 4 D L gcorg cogtit cog IInd 16 10 10,Chapters Pages.
1 The Point 1,2 The Straight Line 44,3 Two or More Straight Lines 117. 4 Change of Axes 169,5 The Circle 179,6 Two or More Circles 219. 7 The Parabola 267,8 The Ellipse 336,9 The Hyperbola 427. 10 The General Equation of the Second Degree and Tracing of Conics 495. 11 System of Conics 546,12 Confocal Conics 559,13 Polar Equation of a Conic 574. 4 D L gcorg cogtit cog IInd 16 10 10,PREFACE TO THE SECOND EDITION.
This book has been my aim to lay before the students a strictly. rigorous and yet very simple exposition of Co ordinate Geometry and its. applications Like the companion volumes Differential Calculus Real. Analysis Differential Equations Integral Calculus Statics Dynamics. etc The book has the following special features,1 The development of the subject is systematics. 2 Almost every article is followed by properly graded illustrative. 3 It contains solutions of questions set in various Indian Universities. and competitive examinations Most of the examples are selected. from the university papers, Errors or misprints if any are unintentional and regretted. Suggestions for further improvement will be warmly received. 4 D L gcorg cogtit cog IInd 16 10 10,DO NOT MISS TO READ. MATHEMATICS SERIES,H HIGHER TRIGONOMETRY,H DEFINITIONS AND FORMULAE IN MATHS. H DIFFERENTIAL EQUATIONS,H DIFFERENTIAL CALCULUS,H DYNAMICS.
H INTEGRAL CALCULUS,H MATRIX ALGEBRA,H MODERN ALGEBRA. H NUMERICAL ANALYSIS,H REAL ANALYSIS,H SEQUENCES AND SERIES. H SOLID GEOMETRY,H THEORY OF NUMBERS,H VECTOR ALGEBRA. H VECTOR CALCULUS,FOR ALL COLLEGE CLASSES,PLEASE WRITE FOR FREE CATALOGUE. 4 D L gcorg cogtit cog IInd 16 10 10,LIST OF SYMBOLS AND ABBREVIATIONS.
Symbol Meaning,belongs to or is an element of,does not belong to. Is a sub set of,Is not a sub set of,Is a super set of. Union of sets,Intersection of sets,A B Cartesian or cross product of sets A and B. U or X Universal set,Ac Complement of A, A B Difference of two sets A and B or complement of B w r t A. Null empty or void set,or or s t such that,iff if and only if.
there exists,implies and is implied by or iff,A B Symmetric difference of A and B. N the set of all natural numbers or positive integers. Z the set of all integers,Q the set of all rational numbers. R the set of all real numbers,is less than or equal to. is greater than or equal to, Sup S or l u b S supermum or least upper bound of S. Inf S or g l b s infimum or greatest lower bound of S. x absolute value of x,a b open interval a x b,a b closed interval a x b.
nbd neighbourhood,A interior of A,A derived set of A. closure of A,f X Y f is a function from X to Y,WP or u infinite series u1 u2 u3. P WP WP alternating series u1 u2 u3 u4,WP QT WP infinite product u1 u2 u3. 4 D L gcorg cogtit cog IInd 16 10 10,1 1 CO ORDINATE GEOMETRY. It is that branch of geometry which defines the position of a point in a plane by a pair of. algebraic numbers It is also called Algebraic Geometry or Analytical Geometry. 1 2 RECTANGULAR AXES AND ORIGIN, Let X OX and Y OY be two perpendicular straight lines Y.
intersecting at the point O Then,i X OX is called the axis of x or the x axis. ii Y OY is called the axis of y or the y axis, iii Both X OX and Y OY taken together in this very order. are called the rectangular axes or the axes of co ordinates 90. or the co ordinate axes or simply the axes X O Origin X. They are called rectangular axes because the angle. between them is a right angle, iv Their point of intersection O is called the origin. 1 3 CARTESIAN CO ORDINATES OF A POINT, Let X OX and Y OY be two perpendicular straight lines intersecting at the point O Let. P be any point in the plane of the axes From P draw PM X OX Then. i OM is called the x coordinate or abscissa of P and is Y. denoted by x P x y, ii MP is called the y coordinate or ordinate of P and is.
denoted by y, iii The numbers x and y are called the Cartesian y. Rectangular Co ordinates or simply the co ordinates of P. iv The symbol P x y is used to denote the point P In this. symbolic representation the abscissa is always written first and. separated from the ordinate by a comma, 1 Abscissa is the distance i e perpendicular distance of a point from y axis. 2 Ordinate is the distance of a point from x axis, 3 Abscissa is ve to the right of y axis and ve to the left of y axis. 4 Ordinate is ve above x axis and negative below x axis. 5 Abscissa of any point on y axis is zero,6 Ordinate of any point on x axis is zero. 7 Co ordinates of the origin are 0 0,2 GOLDEN CO ORDINATE GEOMETRY.
DISTANCE FORMULA, Article 1 1 To Find the Distance Between Two Points Whose Co ordinates are Given. Let P x1 y1 and Q x2 y2 be the given points Draw PL and QM OX and PR QM. Then PR LM OM OL Y,x2 x1 Q x2 y2,RQ MQ MR MQ LP,In right d PQR P x1 y1. PQ PR RQ 2 Pythagoras Theorem,x2 x1 y2 y1 2, Remember Distance between two points x1 y1 and x2 y2. b KHHGTGPEG QH CDUEKUUCGg b KHHGTGPEG QH QTFKPCVGUg. Corollary Distance of a point x y from the origin 0 0. Example 1 Find the distance between the following pairs of points. i am12 2 am1 am22 2 am2 ii a cos a sin a cos a sin. Sol i Reqd distance CO CO,C O O O O O O,C O O O O Assuming m1 m2. ii Reqd distance C EQU C EQU C UKP C UKP,EQU UKP EQU UKP.
EQU EQU UKP UKP,a EQU a EQU,a UKP 2 a UKP, 4 D L gcorg cog1 1 IInd 13 7 10 IIIrd 31 7 10 IVth 24 1 11 Vth 4 1 12. THE POINT 3, Note 1 When three points are given and it is required to prove that. i they form an isosceles triangle show that two of its sides are equal. ii they form an equilateral triangle show that its three sides are equal. iii they form a right angled triangle show that the sum of the squares of two sides is equal to the. square of the third side, iv they are collinear show that the sum of the distances between two point pairs is equal to the. distance between the third point pair, Note 2 When four points are given and it is required to prove that. i they form a square show that all sides are equal and diagonals are equal. ii they form a rhombus show that all sides are equal and diagonals are unequal. iii they form a rectangle show that the opposite sides are equal and diagonals are also equal. iv they form a parallelogram show that the opposite sides are equal. Example 2 Show that the points 2 3 1 2 and 7 0 are collinear. Note Points are collinear means points lie on the same straight line. Sol Let A 2 3 B 1 2 C 7 0 be the given points,so that AB BC AC.
The points A B C are collinear i e lie on the same straight line. Example 3 Show that the points 0 1 2 1 0 3 and 2 1 are the corners of a. Sol Let A 0 1 B 2 1 C 0 3 D 2 1 be the given points. Thus the four sides AB BC CD DA are equal Hence ABCD is either a square of a. rhombus In a square the two diagonals are equal and in a rhombus they are unequal. Let us therefore calculate the diagonals,Thus AC BD. Hence ABCD is a square, Example 4 Show that the four points 7 3 3 0 0 4 and 4 1 are the vertices of. 4 D L gcorg cog1 1 IInd 13 7 10 IIIrd 31 7 10 IVth 24 1 11 Vth 4 1 12. 4 GOLDEN CO ORDINATE GEOMETRY, Sol Let A 7 3 B 3 0 C 0 4 D 4 1 be the given points. AB BC CD DA, ABCD is either a square or a rhombus In a square the two diagonals are equal and. in a rhombus they are unequal Let us therefore calculate the diagonals. Thus AC BD,Hence ABCD is a rhombus, Example 5 Show that the points 3 2 11 8 8 12 and 0 6 are the vertices of a.
Sol Let A 3 2 B 11 8 C 8 12 and D 0 6 be the given points. Thus AB CD and BC DA, The opposite sides of quadrilateral ABCD are equal. So ABCD is either a rectangle or a parallelogram In a rectangle two diagonals are. equal and in a parallelogram they are unequal Let us therefore calculate the diagonals. Diagonals AC and BD are also equal,Hence ABCD is a rectangle. Example 6 Show that the points 1 0 0 3 1 3 and 0 0 are the vertices of a. parallelogram, Sol Let A 1 0 B 0 3 C 1 3 and D 0 0 be the given points. 4 D L gcorg cog1 1 IInd 13 7 10 IIIrd 31 7 10 IVth 24 1 11 Vth 4 1 12. THE POINT 5,AB CD and BC DA, The opposite sides of quadrilateral ABCD are equal. Hence ABCD is a parallelogram, Note For another method see examples with Section Formula.
Example 7 Show that the points 2a 4a 2a 6a and 2a C 5a are the vertices of. an equilateral triangle, Sol Let A 2a 4a B 2a 6a C 2a C 5a be the given points. AB C C C C C 2a,BC C C C C C C C 2a,CA C C C C C C C 2a. Hence ABC is an equilateral triangle, Example 8 Two points 0 0 3 form with another point an equilateral triangle. Find that point, Sol Let A 0 0 B 3 be the given points and C x y the third point so that ABC is. an equilateral triangle,AB BC CA or AB2 BC2 CA2,0 3 2 0 2 3 x 2 y 2 x 0 2 y 0 2.
or 12 x2 y2 6x 2 y 12 x2 y2 1,From the first and third members of 1 we get. x2 y2 12 2,From second and third members of 1 we get. x2 y2 6x 2 y 12 x2 y2,or 6x 2 y 12 0,or 2 y 6x 12 6 2 x. or y 2 x 2 x,Putting y 2 x in 2 we get,x2 3 2 x 2 12 or 4x2 12x 12 12 or 4x2 12x 0. x 3x 0 or x x 3 0 x 0 3,When x 0 y 2 x 2 0 2,When x 3 y 2 x 2 3.
Hence the third point is 0 2 or 3, 4 D L gcorg cog1 1 IInd 13 7 10 IIIrd 31 7 10 IVth 24 1 11 Vth 4 1 12. MATHS SERIES CO ORDINATE GEOMETRY By N P BALI FIREWALL MEDIA An Imprint of Laxmi Publications Pvt Ltd BANGALORE CHENNAI COCHIN GUWAHATI HYDERABAD JALANDHAR KOLKATA LUCKNOW MUMBAI RANCHI NEW DELHI BOSTON USA VED 4 D L gcorg cogtit cog IInd 16 10 10 Published by FIREWALL MEDIA An Imprint of Laxmi Publications Pvt Ltd 113 Golden House Daryaganj New Delhi 110002 Phone 011 43 53

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