arihant coordinate geometry

56.39 MB·english

by  sk goyal

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Skills in
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Mathematics for
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Geometry
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Detailed Instructions inside
■ I 0 o 0 J
H^arihant
Dr. SK Goyal
Skills in
Mathematics for
JEE MAIN &
ADVANCED
Coordinate
Geometry
With Sessionwise Theory & Exercises
Dr. SK Goyal
S*Carihant
*
i ARIHANT PRAKASHAN (Series), MEERUT
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^>$<7 Skills in Mathematics for
JEE MAIN & ADVANCED
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arihant
ARIHANT PRAKASHAN (Series), MEERUT
All Rights Reserved
!fi ©AUTHOR
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Skills in Mathematics for f t
JEE MAIN & ADVANCED
PREFACE
IF YOU CONTINUOUSLY PUT YOUR EFFORTS ON AN ASPECT, YOU HAVE VERY
GOOD CHANCE TO GET POSITIVE OUTCOME i.e. SUCCESS
It is a matter of great pride and honour for me to have received such an overwhelming
response to the previous editions of this book from the readers. In a way, this has inspired
me to revise this book thoroughly as per the changed pattern of JEE Main & Advanced. I
have tried to make the contents more relevant as per the needs of students, many topics
have been re-written, a lot of new problems of new types have been added in etcetc. All
possible efforts are made to remove all the printing errors that had crept in previous
editions. The book is now in such a shape that the students would feel at ease while going
through the problems, which will in turn clear their concepts too.
A Summary of changes that have been made in Revised & Enlarged Edition
• Theory has been completely updated so as to accommodate all the changes made in JEE
Syllabus & Pattern in recent years.
• The most important point about this new edition is, now the whole text matter of each
chapter has been divided into small sessions with exercise in each session. In this way the
reader will be able to go through the whole chapter in a systematic way.
• Just after completion of theory, Solved Examples of all JEE types have been given, providing
the students a complete understanding of all the formats of JEE questions & the level of
difficulty of questions generally asked in JEE.
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• Along with exercises given with each session, a complete cumulative exercises have been
given at the end of each chapter so as to give the students complete practice for JEE along
1
with the assessment of knowledge that they have gained with the study of the chapter.
• Last 13 Years questions asked in JEE Main &Adv, IIT-JEE & AIEEE have been covered in all
5
i the chapters.
However I have made the best efforts and put my all Coordinate Geometry teaching
experience in revising this book. Still I am looking forward to get the valuable
I suggestions and criticism from my own fraternity i.e. the fraternity of JEE teachers.
I would also like to motivate the students to send their suggestions or the changes that
they want to be incorporated in this book. All the suggestions given by you all will be
kept in prime focus at the time of next revision of the book.
i
Dr. SK Goyal
ffi-y Skills in Mathematics for
JEE MAIN & ADVANCED
I
CONTENTS
1. COORDINATE SYSTEM AND COORDINATES 1-71
LEARNING PART Session 3
• Section formulae
Session 1
• Centroid of a Triangle
• Introduction
• Incentre
• Coordinate Axes
• Some Standard Results
• Rectangular Cartesian Coordinates
• Area of Triangle
of a Point
• Polar Coordinates of a Point Session 4
• Relation between the Polar and Cartesian • Locus and Its Equation
Coordinates • Change of Axes the Traisformations
l
Session 2 of Axes
• Distance between Two Points • Removal of the Term x from F(x,y) = ax1 +
• Choice of Axes 2hxy + by1 without Chaiging the Origin
• Distance between Two Points in Polar PRACTICE PART
Coordinates • JEE Type Examples
• Chapter Exercises
2. THE STRAIGHT LINES 73-190
LEARNING PART • The Distance Form or Smmetric Form or
Parametric Form of a Lie
Session 1
• Definition Session 2
• Angle of Inclination of a Line • Position of Two Points Rlative to a
• Slope or Gradient of a Line Given Line
• Angle Between Two Lines
• Position of a Point whichies Inside a Triangle
• Lines Parallel to Coordinate Axes
• Equations of Lines Parall and Perpendicular
• Intercepts of a Line on Axes
to a Given Line
• Different Forms of the Equation of a Straight
• Distance Between Two P allel Lines
Line
• Distance of a Point From Line
• Reduction of General Equation to Standard
• Area of Parallelogram
Form
Skills in Mathematics for
/MA JEE MAIN & ADVANCED
Session 3 Session 5
• Points of Intersection of Two Lines • The Foot of Perpendicular Drawn from the
• Concurrent Lines Point (xp y,) to the Line ax + by + c = 0
• Family of Lines • Image or Reflection of a Point (xp y,)
• How to Find Circumcentre and Orthocentre about a Line Mirror
by Slopes • Image or Reflection of a Point (xp yt) in
Different Cases
Session 4
• Use of Image or Reflection
• Equations of Straight Lines Passing Through a
Given Point and Making a Given Angle with a Session 6
Given Line • Reflection of Light
• A Line Equally Inclined with Two Lines • Refraction of Light
• Equation of the Bisectors • Condition of Collineirty If Three Given Point
• Bisector of the Angle Containing in Cyclic Order
The Origin
PRACTICE PART
• Equation of that Bisector of the Angle Between
• JEE Type Examples
Two Lines Which Contains a Given Point
• Chapter Exercises
• How to Distinguish the Acute (Internal) and
Obtuse (External) Angle Bisectors
3. PAIR OF STRAIGHT LINES 191-239
LEARNING PART Session 5
Session 1 • To Find the Point of Intersection of Lines
• Introduction Represented by ax1 + 2hxy + by2 + 2gx + 2fy +
• Homogeneous Equation in Two Variables c = 0 with the Help of Partial Differentiation
• Removal of First Degree Term
Session 2
• Equation of the Lines Joining the Origin to the
• Angle between the Pair of Lines ax^hxy+by2
Points of Intersection of a Given Line and a
Session 3 Given Curve
• Bisectors of the Angle between the Lines Given
PRACTICE PART
by a Homogeneous Equation
• JEE Type Examples
Session 4 • Chapter Exercises
• General Equation of Second Degree
• Important Theorems
■■'V
Skills in Mathematics for
JEE MAIN & ADVANCED
4. CIRCLE 241-362
LEARNING PART Session 5
Session 1 • Tangents from a Point to the Circle
• Length of the Tangent from a Point to a Circle
• Definition
• Equation of Circles in Different Forms • Power of a Point with Respect to a Circle
• Locus of the Mid-point of the Chords of the • Chord of Contact
Circle that Subtends an Angle of 2q at its Centre • Chord Bisected at a Given Point
• Pair of Tangents
Session 2
• Director Circle
• Diametric Form of a Circle
• Equation of Circle Passing Through Three Session 6
Non-Collinear Points • Diameter of a Circle
• Two Circles Touching Each Other
Session 3
• Common Tangents to Two Circles
• Intercepts Made on the Axes by a Circle
• Common Chord of Two Circles
• Different Forms of the Equations of a Circle
• Family of Circles
• Position of a Point with Respect to Circle
• Maximum and Minimum Distance of a Point Session 7
from the Circle • Angle of Intersection of Two Circles
• Radical Axis
j Session 4
• Radical Centre
• Intersection of a Line and a Circle
• Co-axial System of Circles
• Product of the Algebraical Distances PA and
• Limiting Point
PB is Constant when from P, A Secant be
• Image of the Circle by the Line Mirror
Drawn to Cut the circle in the Point A and B
• The Length of Intercept Cut-off from a Line by PRACTICE PART
a Circle • JEE Type Examples
• Tangent to a Circle at a Given Point • Chapter Exercises
• Normal to a Circle at a Given Point
5. PARABOLA 363-459
LEARNING PART • Standard Equation of Parabola
• Some Terms Related to Parabola
Session 1
• Other forms of Parabola with
• Introduction
• Conic Section Latusrectum 4a
• Section of a Right Circular Cone by Different • Smart Table
Planes • General Equation of a Parabola
• Conic Section : Definition • Equation of Parabola if Equation of
• Equation of Conic Section axis, Tangent at Vertex and Latusrectum are
• Recognisation of Conics given 2
• How to Find the Centre of Conics • The Generalised form (y-fc) = 4a (x-h)
• Parabola: Definition • Parabolic Curve
Skills in Mathematics for
JEE MAIN & ADVANCED
Session 2 Session 3
• Pair of Tangents SSI = T?
• Position of a Point (xpy) with respect to a
Parabola / = 4ax • Chord of Contact
• Parametric Relation between the Coordinates • Equation of the Chord Bisected at a
of the Ends of a Focal Chord of a Parabola Given Point
• Intersection of a Line and a Parabola • Diameter
• Equation of Tangent in Different Forms • Lengths of Tangent, Subtangent, Normal and
• Point of Intersection of Tangents at any Two Subnormal
Points on the Parabola • Some Standard Properties of the Parabola
• Equation of Normals in Different Forms • Reflection Property of a Parabola
• Study of Parabola of the Form (ax + by)^2 + 2gx
• Point of Intersection of Normals at any Two
Points on the Parabola + 2fy + c = 0
• Relation Between 7/ andif Normal at 7/’
PRACTICE PART
meets the Parabola Again at
• JEE Type Examples
• Co-normal Points
• Chapter Exercises
• Circle Through Co-normal Points
..
6. ELLIPSE 461-553
LEARNING PART Co-normal Points Lie on a Fixed Curve
Session 1 Smart Table
• Ellipse Definition Session 3
• Standard Equation of Ellipse • Pair of Tangents
• The Foci and Two Directrices of an • Chord of Contact
Ellipse • Chord Bisected at a Given Point
• Tracing of the Ellipse • Diameter
• Some Terms Related to an Ellipse • Conjugate Diameters
• Focal Distances of a Point 2 . • Properties of Conjugate Diameters
• The Shape of the Ellipse, —— + b2= 1 > • Equi-Conjugate Diameters
• when b>a • Director Circle
• Mechanical Construction of an Ellipse • Sub-Tangent and Sub-Normal
• Smart Table • Concyclic Points
• Some Standard Properties of the Ellipse
Session 2
• Reflection Property of an Ellipse
• Position of a Point with Respect to an Ellipse
• Equation of an Ellipse Referred to Two
• Intersection of a Line and an Ellipse
Perpendicular Lines
• Equation of Tangent in Different Forms
• Equations of Normals in Different Forms PRACTICE PART
• Properties of Eccentric Angles of the Co­ • JEE Type Examples
normal Points • Chapter Exercises
:b.-
/tv-7 Skills in Mathematics for
4A JEE MAIN & ADVANCED
7. HYPERBOLA 555-638
LEARNING PART Session 3
Session 1 • Diameter
• Hyperbola: Definition • Conjugate Diameters
• Standard Equation of Hyperbola • Properties of Hyperbola
• The Foci and Two Directrices of a Hyperbola • Intersection of Conjugate Diameters and
• Tracing of the Hyperbola Hyperbola
• Some Terms Related to Hyperbola • Director Circle
• Focal Distances of a Point • Asymptotes
• Conjugate Hyperbola • Rectangular Hyperbola
2
• Position of a Point with Respect to a Hyperbola • The Rectangular Hyperbola xy = c
2
• Intersection of a Line and a Hyperbola • Study of Hyperbola xy = c
• Properties of Rectangular Hyperbola xy=
Session 2
• Reflection Property of a Hyperbola
• Equations of Tangents in Different Forms
• Equation of a Hyperbola Referred to Two
• Equations of Normals in Different Forms
Perpendicular Lines
• Pair of Tangents
• Chord of Contact PRACTICE PART
• Equation of the Chord Bisected at a • JEE Type Examples
Given Point • Chapter Exercises
T
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V7 Skills in Mathematics for
& JEE MAIN & ADVANCED
SYLLABUS FOR JEE MAIN
Cartesian system of rectangular coordinates in a plane, distance formula, section formula,
locus and its equation, translation of axes, slope of a line, parallel and perpendicular lines,
intercepts of a line on the coordinate axes.
Straight Lines
Various forms of equations of a line, intersection of lines, angles between two lines, conditions
for concurrence of three lines, distance of a point from a line, equations of internal and
external bisectors of angles between two lines, coordinates of centroid, orthocentre and
circumcentre of a triangle, equation of family of lines passing through the point of intersection
of two lines.
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Circles, Conic Sections
Standard form of equation of a circle, general form of the equation of a circle, its radius and
centre, equation of a circle when the end points of a diameter are given, points of intersection
of a line and a circle with the centre at the origin and condition for a line to be tangent to a
circle, equation of the tangent. Sections of cones, equations of conic sections (parabola, ellipse
and hyperbola) in standard forms, condition fory=mx + c to be a tangent and point (s) of
tangency.
SYLLABUS FOR JEE ADVANCED
Analytical Geometry
Two Dimensions Cartesian Coordinates, distance between two points, section formulae, shift
of origin.
Equation of a straight line in various forms, angle between two lines, distance of a point from a
line. Lines through the point of intersection of two given lines, equation of the bisector of the
angle between two lines, concurrency of lines, centroid, orthocentre, incentre and
circumcentre of a triangle.
Equation of a circle in various forms, equations of tangent, normal and chord.
Parametric equations of a circle, intersection of a circle with a straight line or a circle, equation
of a circle through the points of intersection of two circles and those of a circle and a straight
line.
Equations of a parabola, ellipse and hyperbola in standard form, their foci, directrices and
eccentricity, parametric equations, equations of tangent and normal.