JEE Main & Advanced Mathematics Locus

Locus

Category : JEE Main & Advanced

The curve described by a point which moves under given condition or conditions is called its locus.

Equation to the locus of a point : The equation to the locus of a point is the relation, which is satisfied by the coordinates of every point on the locus of the point.

Algorithm to find the locus of a point

Step I : Assume the coordinates of the point say \[(h,\,\,k)\] whose locus is to be found.

Step II : Write the given condition in mathematical form involving \[h,\,\,k\].

Step III : Eliminate the variable (s), if any.

Step IV : Replace \[h\] by \[x\] and \[k\] by \[y\] in the result obtained in step III. The equation so obtained is the locus of the point which moves under some stated condition (s).

Angle Between Two Planes

Equation of Planes Bisecting Angle Between Two Given Planes

Other Topics

Necessary Condition for Extreme Values

Definition

Definition

Length of Perpendicular from Origin to the Tangent

Length of Intercept Made on Axes by The Tangent

Length of Tangent, Normal, Subtangent, Subnormal

Angle of Intersection of Two Curves

Equation of the Tangent and Normal

Slope of the Tangent and Normal

Velocity and Acceleration in Rectilinear Motion

Derivative as the Rate of Change

Deduction of Euler?s Theorem

Euler's Theorem on Homogeneous Functions

Higher Partial Derivatives

Definition

Leibnitz?s Theorem

Differentiation of Integral Function

\[{{n}^{th}}\] Derivative Using Partial Fractions

Successive Differentiation or Higher Order Derivatives

Differentiation of a Function with Respect to Another Function

Methods of Differentiation

Theorems for Differentiation

Some Standard Differentiation

Introduction

Relation Between A.P., G.P. and H.P.

Properties of Arithmetic, Geometric, Harmonic Means Between Two Given Numbers

Special Series

Method of Difference

Method for Finding Sum

Sum of A.G.P.

\[{{n}^{th}}\]term of A.G.P.

Definition

Properties of H.P.

Harmonic Mean

General Term of an H.P.

Definition

Properties of G.P.

Geometric Mean

Sum of infinite terms of a G.P.

Sum of first 'n' terms of a G.P.

Selection of Terms in a G.P.

General term of a G.P.

Definition

Properties of A.P.

Arithmetic Mean

Sum of n terms of an A.P.

Selection of Terms in an A.P.

General Term of an A.P.

Definition

Some Important Results

Some Properties Related to Circle

Some Terminology Related to Height and Distance

Solutions of Triangles

Regular Polygon

Cyclic Quadrilateral

Ex-central Triangle

Pedal Triangle

Circle Connected With Triangle

Half Angle Formulae

Area of Triangle

Napier's Analogy (Law of Tangents)

Theorem of The Medians: (Apollonius Theorem)

Projection Formulae

The Law of Cosines or Cosine Rule

Relation Between Sides and Angles

Periodic Functions

Important Points to be Taken in Case of While Solving Trigonometrical Equations

Method for Finding Principal Value

General Solution of The Form \[\mathbf{acos}\,\mathbf{\theta +bsin}\,\mathbf{\theta =c}\]

Trigonometrical Equations With Their General Solution

Trigonometrical Equations and Inequations Definition

Differentiability of a Function at a Point

Discontinuous Function

Continuity From Left and Right

Continuity of a Function at a Point

Introduction

Methods of Evaluation of Limits

Fundamental Theorems on Limits

Limit of a Function

Inverse Function

Composite Function

Periodic Function

Even and Odd Function

Conditional Trigonometrical Identities

Kinds of function

Algebra of Functions

Domain, Co-domain and Range of Function

Definition of Function

Intervals

Maximum and Minimum Value of a \[\mathbf{cos}\,\,\mathbf{\theta }\,\,\mathbf{+}\,\mathbf{b}\,\,\mathbf{sin}\,\,\mathbf{\theta }\]

Some Important Definitions

Trigonometric Ratio of Sub-multiple of an Angle

Trigonometric Ratio of Multiple of an Angle

Formulae to Rransform The Product Into Sum or Difference

Trigonometrical Ratios In Terms of Each Other

Trigonometrical Ratios for Some Special Angles

Trigonometrical Ratios for Various Angles

Trigonometrical Ratios of Allied Angles

Trigonometrical Ratios or Functions

Domain and Range of a Trigonometrical Function

Angle of Intersection of Two Spheres

Intersection of Straight Line and a Sphere

Condition of Tangency of a Plane to a Sphere

Relation Between an arc and an Angle

Relation Between Three Systems of Measurement of an Angle

System of Measurement of Angles

Section of a Sphere by a Plane

Equation in Sphere in Various Forms

General Equation of Sphere

Projection of a Line on a Plane

Angle Between Line and Plane

Intersection Point of a Line and Plane

Introduction

Equation of Plane Through a Given Line

Coplanar Lines

Image of a Point in a Plane

Equation of Planes Bisecting Angle Between Two Given Planes

Locus

Angle Between Two Planes

Foot of Perpendicular from a Point \[\mathbf{A(\alpha ,}\,\,\mathbf{\beta ,}\,\,\mathbf{\gamma )}\]to a given plane \[\mathbf{ax+by+cz+d=0}\].

Transformation of Axes

Equation of Plane Passing Through the Given Point

Area of Some Geometrical Figures

Definition of Plane and Its Equations

Shortest Distance Between two Straight Lines

Foot of Perpendicular from a Point to The Line

Intersection of Two Lines

Changing Unsymmetrical form to Symmetrical Form

Some Points of a Triangle

Straight Line in Space

Angle Between Two Lines

Projection

Direction Cosines and Direction Ratios

Triangle and Tetrahedron

Section Formulae

Section Formula

Distance Formula

Properties of Some Geometrical Figures

Co-ordinates of a Point in Space

Distance Formula

Polar Co-ordinates

Cartesian Co-ordinates of a Point

Introduction

Binomial Distribution

Total Probability and Baye's Rule

Conditional Probability

Addition Theorems on Probability

Odds in Favour and Odds Against an Event

Problems Based on Combination and Permutation

Classical Definition of Probability

Definitions of Various Terms

Probability Theory

Distance Between The Pair of Parallel Straight Lines

Removal of the term \[\mathbf{xy}\] from \[\mathbf{f(X,}\,\mathbf{Y)=a}{{\mathbf{x}}^{\mathbf{2}}}\mathbf{+2hxy+b}{{\mathbf{y}}^{\mathbf{2}}}\] Without Changing the Origin

Removal of First Degree Terms

Equation of the Lines Joining the Origin to the Points of Intersection of a Given Line and a Given Curve

Bisectors of the Angles Between the Lines

Angle Between the Pair of Lines

Point of Intersection of Lines Represented by \[\mathbf{a}{{\mathbf{x}}^{\mathbf{2}}}\mathbf{+ 2hxy+b}{{\mathbf{y}}^{\mathbf{2}}}\mathbf{+ 2gx+ 2fy+c= 0}\]

Equation of Pair of Straight Lines

Logic Gates

Switching Circuits

Boolean Functions

Introduction

Duality

Algebra of Statements

Tautologies and Contradictions

Negation of Compound Statements

Logical Equivalence

Basic Logical Connectives or Logical Operators

Truth Tables

Types of Statements

Use of Venn Diagrams in Checking Truth and Falsity of Statements

Statements or Propositions

Introduction

Advantages and Limitations of L.P.P.

To Find The Vertices of Simple Feasible Region Without Drawing a Graph

Graphical Solution of Two Variable Linear Programming Problem

Mathematical Formulation of a Linear Programming Problem

Terms of Linear Programming

Linear Inequations

Inverse Trigonometric Ratios of Multiple Angles

Formulae for Sum, Difference of Inverse Trigonometric Function

Properties of inverse trigonometric functions

Domain and Range of Inverse Trigonometric Functions

Graphs of Inverse Trigonometric Functions

Integration of Hyperbolic Functions

Integration of Trigonometric Functions

Integration of Rational Functions by Using Partial Fractions

Integrals of the form \[\int{\frac{a sinx+b cosx}{c sinx+d cosx} }\]and \[\int{\frac{a\,sinx+bcos\,x+q}{c\,sinx+d\,cosx+r}}\]

Integrals of the form \[\int_{{}}^{{}}{\frac{dx}{a+b\mathbf{co}{{\mathbf{s}}^{\mathbf{2}}}x},\,\int_{{}}^{{}}{\frac{dx}{a+b\,\mathbf{si}{{\mathbf{n}}^{\mathbf{2}}}x},}}\]\[\int_{{}}^{{}}{\frac{dx}{a\,\mathbf{si}{{\mathbf{n}}^{\mathbf{2}}}x+b\,\mathbf{co}{{\mathbf{s}}^{\mathbf{2}}}x}\mathbf{,}}\]\[\int_{{}}^{{}}{\frac{dx}{{{\mathbf{(}a\,\sin x+b\,\mathbf{cos}\,x\mathbf{)}}^{\mathbf{2}}}}\mathbf{,}}\int_{{}}^{{}}{\frac{dx}{a\,+b\mathbf{si}{{\mathbf{n}}^{\mathbf{2}}}x+c\,\mathbf{co}{{\mathbf{s}}^{\mathbf{2}}}x}}\]

Integrals of the Form \[\int{\frac{dx}{a+bcosx+csinx}}\,,\,\,\int{\frac{dx}{asinx+bcosx}}\]

Integrals of the Form \[\int{\frac{dx}{a+b\,cos\,x}}\] and \[\int{\frac{dx}{a+b\,sin\,x}}\]

Evaluation of the Various Forms of Integrals by Use of Standard Results

Position of Centre of Gravity in Some Special Cases

Integration by Parts

Centre of Gravity of a Compound Body and Remainder

Centre of Gravity

Limiting Equilibrium on an Inclined Plane

Coefficient of Friction

Friction

Equilibrium of Coplanar Forces

Triangle Theorem of Couples

Integration By Substitution

Couples

Moment

Parallel Forces

Lami's Theorem

Polygon Law of Forces

Triangle Law of Forces

Parallelogram Law of Forces

Fundamental Integration Formulae

Introduction

Definition

Image of a Point in Different Cases

Reflection on The Surface

Concurrent Lines

Position of Two Points with Respect to a Line

Position of a Point with Respect to a Line

Length of Perpendicular

Equations of the Bisectors of The Angles Between two Straight Lines

Equation of Straight Line Through a Given Point Making a Given Angle With a Given Line

Angle Between two Non-parallel Lines

General Equation of Lines Through the Intersection of Two Given Lines

Point of Intersection of Two Lines

General Equation of a Straight Line and its Transformation in Standard Forms

Equation of Parallel and Perpendicular Lines to a Given Line

Equations of Straight Line in Different Forms

Slope (Gradient) of a Line

Definition

Miscellaneous Differential Equation

Application of Differential Equation

Linear Differential Equation

Solution by Inspection

Exact Differential Equation

Homogeneous Differential Equation

Variable Separable Type Differential Equation

Formation of Differential Equation

Definition

Symmetrical Area

Application of Vectors in 3-dimensional Geometry

Rotation of a Vector About an Axis

Vector Product of Four Vectors

Area of Bounded Regions

Scalar Product of Four Vectors

Vector Triple Product

Integration of Piecewise Continuous Functions

Scalar Triple Product

Volumes and Surfaces of Solids of Revolution

Some Important Results of Definite Integral

Area Between Two Curves

Vector or Cross Product

Scalar or Dot Product

Linear Independence and Dependence of Vectors

Linear Combination of Vectors

Position Vector

Leibnitz's Rule

Properties of Vectors

Walli's Formula

Types of Vector

Reduction Formulae for Definite Integration

Gamma Function

Summation of Series by Integration

Representation of Vectors

Introduction

Properties of Definite Integral

Evaluation of Definite Integral by Substitution

Definition

Rectangular or Equilateral Hyperbola

Asymptotes of a Hyperbola

Subtangent and Subnormal of the Hyperbola

Diameter of The Hyperbola

Pole and Polar

Equation of The Chord Joining Two Points on The Hyperbola

Equation of the Chord of the Hyperbola Whose mid point \[\mathbf{(}{{\mathbf{x}}_{\mathbf{1}}}\mathbf{,}\,\,{{\mathbf{y}}_{\mathbf{1}}}\mathbf{)}\] is given

Equation of Chord of Contact of Tangents Drawn from a Point to a Hyperbola

Equations of Normal in Different Forms

Equation of Pair of Tangents

Equations of Tangent in Different Forms

Intersection of a Line and a Hyperbola

Position of a Point With Respect to a Hyperbola

Parametric Equations of Hyperbola

Auxiliary Circle of Hyperbola

Special form of Hyperbola

Conjugate Hyperbola

Standard Equation of the Hyperbola

Definition

Subtangent and Subnormal

Diameter of The Ellipse

Pole and Polar

Equation of The Chord Joining Two Points on an Ellipse

Equation of Chord With Mid Point \[\mathbf{(}{{\mathbf{x}}_{\mathbf{1}}}\mathbf{,}{{\mathbf{y}}_{\mathbf{1}}}\mathbf{)}\]

Chord of Contact

Auxiliary Circle

Equations of Normal in Different Forms

Equation of Pair of Tangents \[\mathbf{S}{{\mathbf{S}}_{\mathbf{1}}}\mathbf{=}{{\mathbf{T}}^{\mathbf{2}}}\]

Equations of Tangent in Different Forms

Intersection of a Line and an Ellipse

Position of a Point with Respect to an Ellipse

Special Forms of an Ellipse

Parametric Form of The Ellipse

Standard Equation of the Ellipse

Definition

Pole and Polar

Length of Tangent, Subtangent, Normal and Subnormal to \[{{\mathbf{y}}^{\mathbf{2}}}\mathbf{= 4ax}\,\,\mathbf{at }\,\mathbf{(a}{{\mathbf{t}}^{\mathbf{2}}}\mathbf{,2at)}\]

Length of Tangent, Subtangent, Normal, Subnormal

Diameter of a Parabola

Equation of the Chord Joining any Two Points on the Parabola

Equation of the Chord of the Parabola Which is Bisected at a Given Point

Equation of The Chord of Contact of Tangents to a Parabola

Co-normal Points

Relation Between \[\mathbf{'}{{\mathbf{t}}_{\mathbf{1}}}\mathbf{'}\] and \[\mathbf{'}{{\mathbf{t}}_{\mathbf{2}}}\mathbf{'}\] if Normal at \[\mathbf{'}{{\mathbf{t}}_{\mathbf{1}}}\mathbf{'}\] Meets the Parabola Again at \[\mathbf{'}{{\mathbf{t}}_{\mathbf{2}}}\mathbf{'}\]

Point of Intersection of Normals at Any Two Points on The Parabola

Equations of Normal in Different Forms

Equation of Pair of Tangents From a Point to a Parabola

Point of Intersection of Tangents at Any Two Points on The Parabola

Equations of Tangent in Different Forms

Position of a Point and a Line With Respect to a Parabola

Parametric Equations of a Parabola

Special form of Parabola \[{{\mathbf{(yk)}}^{\mathbf{2~}}}\mathbf{= 4a}\,\,\mathbf{(xh)= a}\]

Standard Equation of The Parabola

Definition

Recognisation of Conics

General Equation of a Conic Section When Its Focus, Directrix and Eccentricity are Given

Definitions of Various Important Terms

Definition

Image of The Circle by The Line Mirror

Limiting Points

Co-axial System of Circles

Radical Centre

Radical Axis

Family of Circles

Angle of Intersection of Two Circles

Common Tangents to Two Circles

Pole and Polar

Diameter of a Circle

Common Chord of Two Circles

Chord of Contact of Tangents

Normal to a Circle at a Given Point

Power of Point with Respect to a Circle

Director Circle

Pair of Tangents

Length of Tangent

Tangent to a Circle at a Given Point

Intersection of a Line and a Circle

Position of a Point With Respect to a Circle

Intercepts on the Axes

Equation of a Circle in Some Special Cases

Standard Forms of Equation of a Circle

Definition

Divisibility Problems

Second Principle of Mathematical Induction

First Principle of Mathematical Induction

Some Important Points

Three / Four Consecutive Terms or Coefficients

Binomial Theorem for any Index

Multinomial Theorem (for Positive Integral Index)

An Important Theorem

Use of Differentiation and Integration in Binomial Theorem

Properties of Binomial Coefficients

Greatest Term and Greatest Coefficient

To Determine a Particular Term in the Expansion

Middle Term

Number of Terms In The Expansion of \[{{(a+b+c)}^{n}}\] and \[{{(a+b+c+d)}^{n}}\]

Independent Term or Constant Term

General Term

Some Important Expansions

Binomial Theorem for Positive Integral Index

Binomial Expression

Number of Divisors

Multinomial Theorem

Some Important Results For Geometrical Problems

Derangement

Division Into Groups

Conditional Combinations

Number of Combinations With Repetition and All Possible Selections

Number of Combinations Without Repetition

Definition

Circular Permutations

Conditional Permutations

Number of permutations with repetition

Number of Permutations Without Repetition

Definition of Permutation

Introduction

Some Important Result

Equations Which Can Be Reduced to Linear, Quadratic and Biquadratic Equations

Rational Algebraic Inequations

Descarte's Rule of Signs

Position of Roots

Wavy Curve Method

The Quadratic Expression

Properties of Quadratic Equation

Condition for Common Roots

Higher Degree Equations

Relations Between Roots and Coefficients

Nature of Roots

Solution of Quadratic Equation

Types of Quadratic Equation

Polynomial

Composition of Relations

Equivalence Classes of An Equivalence Relation

Types of Relations

Inverse Relation

Definition

Cartesian Product of Sets

Laws of Algebra of Sets

Some important results on number of elements in sets

Operations on sets

Venn-Euler Diagrams

Types of Sets

Introduction