Skills in Mathematics - Coordinate Geometry for JEE Main and Advanced

638 Pages·2021·8.07 MB·english

by  Dr. S K Goyal

Coordinate
Geometry
With Sessionwise Theory & Exercises

Coordinate
Geometry
With Sessionwise Theory & Exercises
Dr. SK Goyal
ARIHANT PRAKASHAN (Series), MEERUT
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P R E FAC E
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.
— 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 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 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 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.
Dr. SK Goyal
CONTENTS
1. COORDINATE SYSTEM AND COORDINATES 1-71
LEARNING PART Session 3
Session 1 — Section formulae
— Introduction — Centroid of a Triangle
— Coordinate Axes — Incentre
— Rectangular Cartesian Coordinates — Some Standard Results
of a Point — Area of Triangle
— Polar Coordinates of a Point
Session 4
— Relation between the Polar and Cartesian
— Locus and Its Equation
Coordinates
— Change of Axes the Transformations of Axes
Session 2
— Removal of the Term xy from F(x,y) = ax2 +
— Distance between Two Points 2hxy + by2 without Changing the Origin
— Choice of Axes
PRACTICE PART
— Distance between Two Points in Polar
Coordinates — JEE Type Examples
— Chapter Exercises
2. THE STRAIGHT LINES 73-190
LEARNING PART — The Distance Form or Symmetric Form or
Session 1 Parametric Form of a Line
— Definition
Session 2
— Angle of Inclination of a Line
— Position of Two Points Relative to a
— Slope or Gradient of a Line Given Line
— Angle Between Two Lines — Position of a Point which lies Inside a Triangle
— Lines Parallel to Coordinate Axes — Equations of Lines Parallel and Perpendicular
— Intercepts of a Line on Axes to a Given Line
— Different Forms of the Equation of a — Distance Between Two Parallel Lines
Straight Line — Distance of a Point From a Line
— Reduction of General Equation to — Area of Parallelogram
Standard Form
Session 3 Session 5
— Points of Intersection of Two Lines — The Foot of Perpendicular Drawn from the
— Concurrent Lines Point (x, y) to the Line ax + by + c = 0
1 1
— Family of Lines — Image or Reflection of a Point (x, y)
1 1
about a Line Mirror
— How to Find Circumcentre and Orthocentre by
Slopes — Image or Reflection of a Point (x, y) in
1 1
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 Points
— Bisector of the Angle Containing in Cyclic Order
The Origin
PRACTICE PART
— Equation of that Bisector of the Angle Between
Two Lines Which Contains a Given Point — JEE Type Examples
— How to Distinguish the Acute (Internal) and — Chapter Exercises
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 ax2 + 2hxy + by2 + 2gx + 2fy +
c = 0 with the Help of Partial Differentiation
— Homogeneous Equation in Two Variables
— Removal of First Degree Term
Session 2
— Equation of the Lines Joining the Origin to the
— Angle between the Pair of Lines ax2+2hxy+by2
Points of Intersection of a Given Line and a
Session 3 Given Curve
— Bisectors of the Angle between the Lines
PRACTICE PART
Given by a Homogeneous Equation
— JEE Type Examples
Session 4
— Chapter Exercises
— General Equation of Second Degree
— Important Theorems
4. CIRCLE 241-362
LEARNING PART Session 5
Session 1 — Tangents from a Point to the Circle
— Definition — Length of the Tangent from a Point to a Circle
— 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
Session 2 — Pair of Tangents
— Diametric Form of a Circle — Director Circle
— Equation of Circle Passing Through Three Non- Session 6
Collinear Points
— Diameter of a Circle
Session 3 — Two Circles Touching Each Other
— Intercepts Made on the Axes by a Circle — Common Tangents to Two Circles
— Different Forms of the Equations of a Circle — Common Chord of Two Circles
— Position of a Point with Respect to Circle — Family of Circles
— Maximum and Minimum Distance of a Point
Session 7
from the Circle
— Angle of Intersection of Two Circles
Session 4
— Radical Axis
— Intersection of a Line and a Circle — Radical Centre
— Product of the Algebraical Distances PA and — Co-axial System of Circles
PB is Constant when from P, A Secant be
— Limiting Point
Drawn to Cut the circle in the Point A and B
— Image of the Circle by the Line Mirror
— The Length of Intercept Cut-off from a Line by
a Circle PRACTICE PART
— Tangent to a Circle at a Given Point — JEE Type Examples
— Normal to a Circle at a Given Point — Chapter Exercises
5. PARABOLA 363-459
LEARNING PART — Intersection of a Line and a Parabola
Session 1 — Equation of Tangent in Different Forms
— Introduction — Point of Intersection of Tangents at any Two
— Conic Section Points on the Parabola
— Section of a Right Circular Cone by Different — Equation of Normals in Different Forms
Planes — Point of Intersection of Normals at any Two
— Conic Section : Definition Points on the Parabola
— Equation of Conic Section — Relation Between ‘t1’ and ‘t2’ if Normal at ‘t1’
meets the Parabola Again at ‘t’
— Recognisation of Conics 2
— Co-normal Points
— How to Find the Centre of Conics
— Circle Through Co-normal Points
— Parabola : Definition
— Standard Equation of Parabola Session 3
— Some Terms Related to Parabola — Pair of Tangents SS1 = T2
— Other forms of Parabola with — Chord of Contact
Latusrectum 4a — Equation of the Chord Bisected at a
— Smart Table Given Point
— General Equation of a Parabola — Diameter
— Equation of Parabola if Equation of — Lengths of Tangent, Subtangent, Normal and
axis, Tangent at Vertex and Latusrectum Subnormal
are given — Some Standard Properties of the Parabola
2
— The Generalised form (y-k) = 4a (x-h) — Reflection Property of a Parabola
— Parabolic Curve — Study of Parabola of the Form
2
(ax + by) + 2gx + 2fy + c = 0
Session 2
— Position of a Point (x, y) with respect to a PRACTICE PART
1 1
Parabola y2 = 4ax
— JEE Type Examples
— Parametric Relation between the Coordinates
— Chapter Exercises
of the Ends of a Focal Chord of a Parabola