Aspiring engineers have always wanted to secure admission in pioneering institutes such as the Indian Institute of Technology and the National Institute of Technology. Fundamentals of Mathematics, the complete mathematics collection, prepares students for such prestigious entrance examinations. This series is customized, class-tested and structure-driven with a conceptual approach to the subject. The authority, command and experience of the author, Sanjay Mishra, is reflected in the clear explanations of complex concepts and in the chapter-end exercises. Each volume of this series is meticulously planned in a unique student-friendly manner to make the learning process easier, more effective and enjoyable.
Salient Features * Strictly aligned with the prescribed syllabus. Written in a lucid manner to assist students to understand the concepts without the help of any guide. * Provides the vast subject in a structured and useful manner so as to familiarize the candidates taking the current examinations with the current trends and types of multiple-choice questions asked. * The multiple-choice questions have been arranged in following categories: * Straight Objective Type Questions (Single Choice) * Brainteasers Objective Type Questions (Single Choice) * Multiple Correct Answer Type Questions (More than one choice) * Linked-Comprehension Type Questions * Assertion and Reasoning Questions * Matrix-Match Type Questions and the IIT - JEE Corner
Table of Contents * Preface * Acknowledgements
Chapter 1 Point and Cartesian System * Introduction * Postulates of Euclidean Geometry * Frame of reference * Co-ordinate Systems * Rectangular co-ordinate system * Sign convention * Oblique Co-ordinate System * Polar co-ordinate system * Relation between the polar and cartesian co-ordinates * Distance Between Two Points Lying in a Plane * When co-ordinates of two points are given in rectangular form * When the co-ordinates of point are given in oblique system * When the co-ordinates of points are given in polar form * Application of Distance Formula * Collinearity of three given points * Formation of different triangles * Formation of quadrilaterals * Section Formula (Division of a Line Segment by a Point) * Points of Tri-section * Applications of Section Formula * Area of Geometrical Figures * Area of triangle * Stair method to find the area of n sided polygon * Area of general quadrilateral * Slope of Line Segment * Angle Between Two Line Segments * Standard Points of a Triangle * Centroid * Circum circle and circumcentre * Orthocentre * Incircle and incentre * Ex-circles and ex-centres * Nine Point Circle * Selection of Axes * Geometrical Transformations * Transformation of Axes * Locus * Properties of Equation of Locus * Method to Find Out the Locus * Intersection of Loci * Locus Represented by Combined Equations
Multiple Choice Questions * Objective Type Solved Examples * Subjective Type Solved Examples * Tutorial Exercise * Answer Keys
Chapter 2 Straight Line * Introduction * Definition of Straight Line * General equation of straight line * Equation of straight line parallel to axes * Slope of a Line * Different Forms of the Equation of Straight Line * Slope intercept form * Reduction of general form into slope-intercept form * Slope point form of a line * Two point form equation of a line * Intercept form of a line and concept of line at infinity * Reduction of general form into intercept form (x/a + y/b = 1) * Normal/perpendicular form of a line * Parametric/Symmetric or Distance Form of Line * Oblique Distance of a Point from a Line * Position of Point with Respect to Line * Position of a point with respect to a triangle * Angle Between Two Straight Lines * Some important results * Conditions for two lines to be parallel/coincident/perpendicular * Straight Line throught a Given Point (x1, y1) Making an Angle α with a Given Straight Line y = mx + c * Distance Between Two Parallel Lines * Distance of a Point from a Line * Image of a Point in a Line * Family of Straight Lines * Definition of family of lines * Family of lines passing through intersection of two lines * Area of Parallelogram * Equation of a Reflected Ray in a Mirror * Equation of the Bisectors of the Angles Between Lines * Bisector of Angle Containing the Origin * Whether the Origin Lies in the Obtuse or Acute Angle? * Bisector of Acute and Obtuse Angle * Equation of Bisector of Angle Between Two Lines Containing a Given Point A(α, β) * To Determine if P(α, β) Lies in Acute/Obtuse Angle Between the Two Straight Lines L1 and L2 * Intersection of Two Lines and Condition for Concurrency of Three Lines * Intersection of two lines * Condition for Concurrency of Three Lines * Pair of Straight Lines * Condition for the General Equation to Represent a Pair of Straight Lines and Method to Seperate Them * Point of Intersection of Pair of Straight Lines * Angles Between Pair of Straight Lines * Sufficient Conditions for Pair of Straight Lines ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 to Represent Intersecting/Parallel/Coincident Lines * Homogenous Equation in Two Variables x and y * Pair of Straight Lines Through the Origin * Angle Bisectors of Pair of Straight Lines ax2 + 2hxy + by2 = 0 * Angle Bisectors of Pair of Straight Lines ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 * Isoinclined Pair of Straight Lines * Equation of Pair of Straight Lines Joining Origin to the Point of Intersection of a Curve and a Straight Line
Multiple Choice Questions * Objective Type Solved Examples * Subjective Type Solved Examples * Tutorial Exercise * Answer Keys
Chapter 3 Circles and Family of Circles * Introduction * Review of Basic Geometry of Circles * Definition * Equation of a Circle in Various Forms * Centre radius form/point circle form * General equation * Concentric Circle * Diameteric form * Equation of Circle in Particular Forms * Radius is ‘a’ and touches at (x1, 0) * Radius is ‘a’ and touches y-axis at (0, y1) * Equation of circle touching both axes * Equation of circle when circle passes through the origin and centre lies on x-axis * Equation of circle when circle passes through the origin and centre lies on y-axis * Equation of circle through origin and x-intercept and y-intercept are a and b respectively * The Equation of Circle through Three Non-Collinear Points * Parametric Equation of a Circle * Position of a Point/Line with Respect to a Circle * Maximum and Minimum Distance of a Point from the Circle * Position of Line with Respect to Circle * Image of the Circle in the Line Mirror * Condition for Tangency * Point of Intersection * Length of Intercept * Intercept Made on Co-ordinate Axes by the Circle * Equation of Tangent and Normal * Equation of a tangent at a point on a circle * Calculus method * Parametric form * Equation of normal * Length of the Tangent * Pair of tangent * Director Circle * Chords of Circle * Chord with mid-point (x1, y1) * Chord of Contact * Diameter of a Circle * Relative Position of Two Circles * Case I: When two circles lie outside of each other * Length of direct common tangent * Length of transverse common tangent * Case II: When two circles touch each other externally * Length of direct common tangent * Case III: When two circles intersect each other * Angle of intersection * Case IV: When two circles touch each other internally * Case V: When smaller circle completely lies in the bigger circle * Family of Circle * Radical Axis and Radical Centre * Properties of the radical axis * Radical Centre * Coaxial System (Family) of Circles * Limiting point of a coaxial system * Orthogonal circles of a coaxial system * System of coaxial circles whose two limiting points are given * Properties of limiting points
Multiple Choice Questions * Objective Type Solved Examples * Subjective Type Solved Examples * Tutorial Exercise * Answer Keys
Chapter 4 Parabola Introduction * Conic Section * Circle, Ellipse, Parabola and Hyperbola * Degenerated conic sections * Definition of Conics * Definition of Various Terms Related to Conic * Equation of Conic Section * Parabola and its Related Terms and Properties * Definition * Standard parabola * Four Standard Types of Parabolas and Their Equations and Related Terms * Non-standard Forms of Parabola * Parabola having its vertex at (α, β) is not at origin, axis parallel to x-axis and length of latus rectum ‘4a’ * Parabola having its vertex at (α, β) i.e., not origin, axis parallel to y-axis and length of latus rectum ‘4a’ * Parabola having its axis oblique, vertex at (α, β), and latus rectum 4a * General Equation of Parabola * Parametric Equation of Parabola * Position of Point and Line with Respect to Parabola * Position of Line with Respect to Parabola * Point of intersection * Point of contact * Point of intersection * Chords of Parabola * Chord of parabola in parametric form * Condition for a chord to be a focal chord * Properties of focal chord * Equation of chord whose mid-point is (x1, y1) * Equation of Tangent in Different Forms and Their Properties * Point form * Parametric form * Slope form * Properties of Tangents * Equation of Pair of Tangents * Equations of Normals in Different Forms * Point form * Parametric form * Slope form * Co-normal Points * Properties of Normal and Co-normal Points * Circles through Co-normal Points * Chord of Contact
Multiple Choice Questions * Objective Type Solved Examples * Subjective Type Solved Examples * Tutorial Exercise * Answer Keys
Chapter 5 Ellipse * Introduction * Definition of Ellipse * Standard Ellipse * Equation of standard ellipse * Terms Related to Standard Ellipse * Focal Distance of a Point on Ellipse * Second Definition of Ellipse * Mechanical Construction of an Ellipse * Tracing of the Ellipse * Basic Terminology and Ellipse at a Glance * Non Standard Ellipse with Their Axes Parallel to Co-ordinates Axes and Centre Not at Origin * Non Standard Elllipse with Axes Inclined to Co-ordinate Axes: (Centre May Be at Origin) * Position of a Point with Respect to an Ellipse * Auxiliary Circle and Eccentric Angle * Parametric Equation of the Ellipse * Chord of Ellipse * Focal chord * Intersection of a Line and an Ellipse * Condition of Tangency and Tangent to Ellipse * Point of Contact of Tangent * Equations of Tangent to Ellipse in Different Forms * Pair of Tangents * Director Circle * Chord of Contact * Chord with a Given Mid-point * Properties of Tangents to Ellipse * Different Forms of Normals to Ellipse * Co-normal Points * Properties of Normal to Ellipse and Co-normal Points * Reflection Property of an Ellipse * Con-cyclic Points of Ellipse * Miscellaneous Terms Related to Ellipse * Diameter * Conjugate diameters * Properties of conjugate diameters * Sub-tangent and Sub-normal
Multiple Choice Questions * Objective Type Solved Examples * Subjective Type Solved Examples * Tutorial Exercise * Answer Keys
Chapter 6 Hyperbola * Introduction * Basic Features of Hyperbola * Definition * Standard equation of hyperbola * Tracing of the hyperbola * Some terms related to hyperbola * Second definition of hyperbola * Rectangular hyperbola * Conjugate hyperbola * Hyperbola with their axes || to co-ordinate axis and centre shifted to (α, β) * Equation of hyperbola referred to two perpendicular straight lines as their axes, but not parallel to co-ordinate axes * Hyperbola and basic definitions at a glance * Auxiliary Circle of Hyperbola and Eccentric Angle of Point on Hyperbola * Eccentric Angle * Parametric Equation * Chords of Hyperbola in Cartesian Form * Chord of Hyperbola in Parametric Form * Focal Chord of Hyperbola and its Property * Position of a Point with Respect to Hyperbola * Position of a Straight Line with Respect to Hyperbola * Condition of Tangency of a Straight Line with Respect to Hyperbola * Equation of Tangent to Hyperbola in Cartesian Form * Equation of Tangent to Hyperbola in Parametric Form * Equation of Tangent in Slope Form * Director Circle * Equation of Normal to Hyperbola in Cartesian Form * Equation of Normal to Hyperbola in Parametric Form * Properties of Tangent and Normal * Co-normal Points * Properties of Co-normal Points * Equation of Chord Bisected at a Given Point * Pair of Tangents * Chord of Contact * Diameter of Hyperbola * Conjugate Diameters * Properties of conjugate diameters * Sub Tangent and Sub normal * Asymptotes to a Hyperbola * Properties of Asymptotes to Hyperbola * Rectangular Hyperbola and Their Properties (xy = c2) * Features of Rectangular Hyperbola x2 – y2 = a2 * The retangular Hyperbola with Co-ordinate Axes as Asymtotes * Parametric Form of Rectangular Hyperbola with X-Axis and Y-Axis as Their Asymptotes * Conjugate Hyperbola of Rectangular Hyperbola xy = c2 * Parametric Equations of Chord/Tangent and Normal to Rectangular Hyperbola xy = c2 * Chord of Rectangular Hyperbola xy = c2 with a given Middle Point as (h, k) is kx + hy = 2hk
Multiple Choice QuestionsM * Objective Type Solved Examples * Subjective Type Solved Examples * Tutorial Exercise * Answer Keys
Title
Fundamentals of Mathematics Coordinate Geometry (Paperback)
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