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      Class 11 MATHS – JEE

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      • Class 11
      • Class 11 MATHS – JEE
      CoursesClass 11MathsClass 11 MATHS – JEE
      • 1.Sets, Relation and Functions
        12
        • Lecture1.1
          Introduction to sets, Description of sets 32 min
        • Lecture1.2
          Types of Sets, Subsets 39 min
        • Lecture1.3
          Intervals, Venn Diagrams, Operations on Sets 37 min
        • Lecture1.4
          Laws of Algebra of Sets 26 min
        • Lecture1.5
          Introduction to sets and its types, operations of sets, Venn Diagrams 28 min
        • Lecture1.6
          Functions and its Types 38 min
        • Lecture1.7
          Functions Types 17 min
        • Lecture1.8
          Cartesian Product of Sets, Relation, Domain and Range 40 min
        • Lecture1.9
          Sum Related to Relations 04 min
        • Lecture1.10
          Sums Related to Relations, Domain and Range 22 min
        • Lecture1.11
          Chapter Notes – Sets, Relation and Functions
        • Lecture1.12
          NCERT Solutions – Sets, Relation and Functions
      • 2.Trigonometric Functions
        28
        • Lecture2.1
          Introduction, Some Identities and Some Sums 16 min
        • Lecture2.2
          Some Sums Related to Trigonometry Identities, trigonometry Functions Table and Its Quadrants 35 min
        • Lecture2.3
          NCERT Sums Ex.3.3 (Q.1-5)Based on Trigometry table and Their Quadrants, Trigonometry Identities of Sum and Diff. of two Angles 21 min
        • Lecture2.4
          NCERT Sums Ex-3.2 Based on Trigonometry Function of Lower & Higher Angles 22 min
        • Lecture2.5
          NCERT Sums Ex-3.3 (Q.6 – 10) Based on Radian Angles 11 min
        • Lecture2.6
          NCERT Sums Ex-3.3 (Q.11-13)Based on Trigonometry Identities 16 min
        • Lecture2.7
          NCERT Sums Ex-3.3 (Q. 14)Based on Trigonometry Identities 14 min
        • Lecture2.8
          NCERT Sums Ex-3.3 (Q.16) Based on Trigonometry Identities 05 min
        • Lecture2.9
          NCERT Sums Ex-3.3 (Q.17 -21) Based on Trigonometry Identities 12 min
        • Lecture2.10
          NCERT Sums Ex-3.4 (Q. 1 – 9), Trigonometry Equation 25 min
        • Lecture2.11
          Sums Based on Trigonometry Equations 24 min
        • Lecture2.12
          Sums Based on Trigonometry Equations 11 min
        • Lecture2.13
          Sums Based on Trigonometry Equations 11 min
        • Lecture2.14
          Sums Based on Trigonometry Equations 17 min
        • Lecture2.15
          Equations Having two Variable Angle which satisfy both equations 10 min
        • Lecture2.16
          Trigonometrical Identities-Some important relations and Its related Sums 16 min
        • Lecture2.17
          Sums Related to Trigonometrical Identities 18 min
        • Lecture2.18
          Properties of Triangles and Solution of Triangles-Sine formula, Napier Analogy and Sums 17 min
        • Lecture2.19
          Relation Between Degree and Radian, Quadrant and NCERT Sum Ex.3.1, 3.2 41 min
        • Lecture2.20
          Trigonometric Functions Table 09 min
        • Lecture2.21
          Some Trigonometric Identities and its related Sums 42 min
        • Lecture2.22
          Sums Related to Trigonometrical Identities 19 min
        • Lecture2.23
          Sums Related to Trigonometrical Identities 41 min
        • Lecture2.24
          Sums Related to Trigonometrical Identities 23 min
        • Lecture2.25
          Trigonometry Equations 44 min
        • Lecture2.26
          Sum Based on Trigonometry Equations 07 min
        • Lecture2.27
          Sums Based on Trigonometry function of Lower Angle 03 min
        • Lecture2.28
          Chapter Notes – Trigonometric Functions
      • 3.Mathematical Induction
        5
        • Lecture3.1
          Introduction to PMI 25 min
        • Lecture3.2
          NCERT Solution of EX- 4.1 14 min
        • Lecture3.3
          NCERT Solution of EX- 4.1 22 min
        • Lecture3.4
          NCERT Solution of EX- 4.1 13 min
        • Lecture3.5
          Chapter Notes – Mathematical Induction
      • 4.Complex Numbers and Quadratic Equation
        15
        • Lecture4.1
          Introduction, Nature of Roots, Numbers, Introduction of i 27 min
        • Lecture4.2
          Sum Related to Relations, Real and Imaginary part of C-N, Conjugate of a C-N 33 min
        • Lecture4.3
          Absolute value or Modulus of a C-N and Related Sums 29 min
        • Lecture4.4
          Sums Related To Multiplicative Inverse 29 min
        • Lecture4.5
          Polar Form of a C-N 32 min
        • Lecture4.6
          Sums Related To Polar Form 32 min
        • Lecture4.7
          Square Roots of C-N and its Related Sums 28 min
        • Lecture4.8
          De Moivris Theorem and its related Sums 31 min
        • Lecture4.9
          Introduction, Nature of Roots, Numbers, Introduction of i and its Sums, Real and Imaginary Part of C-N 35 min
        • Lecture4.10
          Sums Related to Real and Imaginary Part of C-N and Operations on C-N 13 min
        • Lecture4.11
          Sums Related To Multiplicative Inverse 07 min
        • Lecture4.12
          Sums Related To Multiplicative Inverse and Modulus and Argument of a C-N 33 min
        • Lecture4.13
          Polar form of a C-N, Nature of Roots 38 min
        • Lecture4.14
          Sums Based on Roots of Quadratic Equations, Sums of Polar form 10 min
        • Lecture4.15
          Chapter Notes – Complex Numbers and Quadratic Equation
      • 5.Linear Inequalities
        4
        • Lecture5.1
          Introduction, Solve some Linear Inequalities and its Graph 42 min
        • Lecture5.2
          Solve some Linear Inequalities and its Graph 12 min
        • Lecture5.3
          Solve some Linear Inequalities and its Graph and Introduction-Permutations and Combinations 35 min
        • Lecture5.4
          Chapter Notes – Linear Inequalities
      • 6.Permutations and Combinations
        5
        • Lecture6.1
          NCERT Sums Ex-7.3, Equation 41 min
        • Lecture6.2
          NCERT Sums Ex-7.3, Equation 02 min
        • Lecture6.3
          Combination and NCERT Sums Ex-7.4 40 min
        • Lecture6.4
          NCERT Sums Ex-7.1 & 7.2 22 min
        • Lecture6.5
          Chapter Notes – Permutations and Combinations
      • 7.Binomial Theorem
        19
        • Lecture7.1
          Introduction to Binomial Theorem 21 min
        • Lecture7.2
          Binomial General Expansion and Their Derivations and its Related Sums 22 min
        • Lecture7.3
          Pascal’s Triangle Theorem, Addition of Two Expansion, NCERT Sums Ex-8.1 26 min
        • Lecture7.4
          Sums of Miscellaneous Exercise and Ex-8.1, Finding the Any Term from nth Term 42 min
        • Lecture7.5
          NCERT Sums Ex-8.1 14 min
        • Lecture7.6
          NCERT Sums Ex-8.1 04 min
        • Lecture7.7
          NCERT Sums Ex-8.2, Middle Term 21 min
        • Lecture7.8
          NCERT Sums Ex-8.2, Middle Term Related Sums 08 min
        • Lecture7.9
          To Find the Coefficient of X^r in the Expansion of (X+A)^n, NCERT Sums Ex-8.2 and Miscellaneous Ex. 40 min
        • Lecture7.10
          NCERT Sums Ex-8.2 10 min
        • Lecture7.11
          To Find the Sum of the Coefficients in the Expansion of (1+x)^n and its Related Sums 27 min
        • Lecture7.12
          Sums Related to Binomials Coefficients 24 min
        • Lecture7.13
          Binomial Theorem for any Index and its Related Sums 27 min
        • Lecture7.14
          Introduction to Binomial Theorem, General Term in the Expansion of (x+a)^n. 39 min
        • Lecture7.15
          NCERT Sums Ex-8.1 & 8.2, Pascals’ Triangle, pth Term from End 24 min
        • Lecture7.16
          Sums related to Finding the Coefficient, NCERT Sums Ex-8.2, Middle Term 40 min
        • Lecture7.17
          Sums Related to Middle Term 17 min
        • Lecture7.18
          Sums Related to Coefficient of the Any Term 31 min
        • Lecture7.19
          Chapter Notes – Binomial Theorem
      • 8.Sequences and Series
        14
        • Lecture8.1
          Introduction, A.P., nth Term and Sum of nth Term, P Arithmetic Mean B/w a and b, Sum Based on Fibonacci Sequence 27 min
        • Lecture8.2
          NCERT Sums Ex-9.2 37 min
        • Lecture8.3
          NCERT Sums Ex-9.2 18 min
        • Lecture8.4
          NCERT Sums Ex-9.2, Geometric Progression -Introduction, nth term, NCERT Sums Ex-9.3 39 min
        • Lecture8.5
          NCERT Sums Ex-9.3 16 min
        • Lecture8.6
          Sum of n term of G.P., NCERT Sums Ex-9.3 40 min
        • Lecture8.7
          NCERT Sums Ex-9.3 08 min
        • Lecture8.8
          NCERT Sums Ex-9.3, Insert P Geometrical Mean B/w a and b 36 min
        • Lecture8.9
          NCERT Sum Ex-9.3 17 min
        • Lecture8.10
          NCERT Sum Ex-9.3 09 min
        • Lecture8.11
          Some Special Series, NCERT Sum Ex-9.4 36 min
        • Lecture8.12
          NCERT Sum Ex-9.4 02 min
        • Lecture8.13
          NCERT Sum Ex-9.4 18 min
        • Lecture8.14
          Chapter Notes – Sequences and Series
      • 9.Properties of Triangles
        2
        • Lecture9.1
          Sine 7 Cosine Rule, Projection Formulae, Napier’s Analogy, Incircle, Some Sums 41 min
        • Lecture9.2
          Angle of Elevations and Depression. and Its Related Sums 13 min
      • 10.Straight Lines
        30
        • Lecture10.1
          Introduction, Equation of Line, Slope or Gradient of a line 24 min
        • Lecture10.2
          Sums Related to Finding the Slope, Angle Between two Lines 22 min
        • Lecture10.3
          Cases for Angle B/w two Lines, Different forms of Line Equation 23 min
        • Lecture10.4
          Sums Related Finding the Equation of Line 27 min
        • Lecture10.5
          Sums based on Previous Concepts of Straight line 32 min
        • Lecture10.6
          Parametric Form of a Straight Line 16 min
        • Lecture10.7
          Sums Related to Parametric Form of a Straight Line 16 min
        • Lecture10.8
          Sums Based on Concurrent of lines, Angle b/w Two Lines 45 min
        • Lecture10.9
          Different condition for Angle b/w two lines 04 min
        • Lecture10.10
          Sums Based on Angle b/w Two Lines 36 min
        • Lecture10.11
          Equation of Straight line Passes Through a Point and Make an Angle with Another Line 09 min
        • Lecture10.12
          Sums Based on Equation of Straight line Passes Through a Point and Make an Angle with Another Line 15 min
        • Lecture10.13
          Sums Based on Equation of Straight line Passes Through a Point and Make an Angle with Another Line 17 min
        • Lecture10.14
          Finding the Distance of a point from the line 34 min
        • Lecture10.15
          Sum Based on Finding the Distance of a point from the line and B/w Two Parallel Lines 33 min
        • Lecture10.16
          Sums Based on Find the Equation of Bisector of Angle Between two intersecting Lines 44 min
        • Lecture10.17
          Sums Based on Find the Equation of Bisector of Angle Between two intersecting Lines 02 min
        • Lecture10.18
          Introduction, Distance B/w Two Points, Slope, Equation of Line 32 min
        • Lecture10.19
          NCERT Sums Ex-10.1 43 min
        • Lecture10.20
          NCERT Sums Ex-10.1 29 min
        • Lecture10.21
          NCERT Sums Ex-10.1 & 10.2 43 min
        • Lecture10.22
          NCERT Sums Ex-10.2 30 min
        • Lecture10.23
          NCERT Sums Ex-10.2 41 min
        • Lecture10.24
          NCERT Sums Ex-10.2 & 10.3 21 min
        • Lecture10.25
          NCERT Sums Ex- 10.3 (Reduce the Equation into intercept Form, Normal form) 42 min
        • Lecture10.26
          NCERT Sums Ex-10.3 21 min
        • Lecture10.27
          NCERT Sums Ex-10.3 (Equation of Parallel line, Perpendicular Line of given line, Sums Based of Angle B/w Two Lines) 42 min
        • Lecture10.28
          NCERT Sums Ex-10.3 09 min
        • Lecture10.29
          NCERT Sums Ex-10.3 26 min
        • Lecture10.30
          Chapter Notes – Straight Lines
      • 11.Conic Sections
        21
        • Lecture11.1
          Introduction, General Equation of second Degree, Parabola, Sums based on Finding Equation of Parabola 41 min
        • Lecture11.2
          Sums Based on Equation of Parabola, Four Forms of Parabola-Form (i) 30 min
        • Lecture11.3
          Sums Based on Four Forms of Parabola-Form (i) 32 min
        • Lecture11.4
          Four Forms of Parabola-Form (ii), (iii) (iv) 13 min
        • Lecture11.5
          Sums Based on Four forms of Parabola 18 min
        • Lecture11.6
          Position of a Point with Respect to Parabola and its Sums 43 min
        • Lecture11.7
          Circles-Introduction, Different Cases for Circle Equations, NCERT Sums Ex-11.1 16 min
        • Lecture11.8
          NCERT Sums Ex-11.1 40 min
        • Lecture11.9
          Circle Important Point Revise, Intersection of Axes, NCERT Sums Ex-11.1 11 min
        • Lecture11.10
          NCERT Sums Ex-11.1 44 min
        • Lecture11.11
          Parabola- Introduction, General Equation , Sums, Some Important Concepts for Parabola 12 min
        • Lecture11.12
          Different Form of Parabola, NCERT Sum Ex-11.2 13 min
        • Lecture11.13
          NCERT Sum Ex-11.2 34 min
        • Lecture11.14
          Ellipse-Introduction, General Equation, NCERT Sums Ex-11.3 36 min
        • Lecture11.15
          NCERT Sums Ex-11.3 02 min
        • Lecture11.16
          NCERT Sums Ex-11.3 23 min
        • Lecture11.17
          Hyperbola-Introduction, NCERT Sums Ex-11.4 12 min
        • Lecture11.18
          NCERT Sums Ex-11.4 25 min
        • Lecture11.19
          Chapter Notes – Conic Sections Circles
        • Lecture11.20
          Chapter Notes – Conic Sections Ellipse
        • Lecture11.21
          Chapter Notes – Conic Sections Parabola
      • 12.Coordinate Geometry
        8
        • Lecture12.1
          Introduction to Rectangular Cartesian Coordinate Geometry (2D), Distance b/w two points 23 min
        • Lecture12.2
          Cartesian Coordinate of points 32 min
        • Lecture12.3
          Questions rel to cartesian coordinate of points 25 min
        • Lecture12.4
          Section Formula – Case 1, Case 2 24 min
        • Lecture12.5
          Problem Solving 26 min
        • Lecture12.6
          Centeroid, Incenter, Circumcenter of a triangle 30 min
        • Lecture12.7
          Locus Problems 17 min
        • Lecture12.8
          Problem Solving 21 min
      • 13.Three Dimensional Geometry
        3
        • Lecture13.1
          Introduction to 3D 18 min
        • Lecture13.2
          Numerical problems 14 min
        • Lecture13.3
          Chapter Notes – Three Dimensional Geometry
      • 14.Limits And Derivatives
        12
        • Lecture14.1
          Introduction to limits 42 min
        • Lecture14.2
          EX-13.1 16 min
        • Lecture14.3
          Questions based on algebra of limits 41 min
        • Lecture14.4
          Limits of a polynomial 12 min
        • Lecture14.5
          rational function 37 min
        • Lecture14.6
          trigo function 21 min
        • Lecture14.7
          Introduction to Derivatives 37 min
        • Lecture14.8
          Ex-13.2 22 min
        • Lecture14.9
          Algebra of derivatives 38 min
        • Lecture14.10
          Derivative of polynomial 13 min
        • Lecture14.11
          trigo function 11 min
        • Lecture14.12
          Chapter Notes – Limits And Derivatives
      • 15.Mathematical Reasoning
        3
        • Lecture15.1
          What is statement ? Special word and phrases, negation of statement , Compound statement , and & or in compound statement , truth table Solving the problems of Ex- 14.1 , 14.2 25 min
        • Lecture15.2
          Solving Ex-14.3, Ex-14.4, Implications, Validating statements, Ex-14.5, Direct method 24 min
        • Lecture15.3
          Chapter Notes – Mathematical Reasoning
      • 16.Statistics
        5
        • Lecture16.1
          Mean, Median, Mode, Range, Mean Deviation Solution of Ex-15.1 27 min
        • Lecture16.2
          Mean Deviation about Mean & Median, Ex-15.2, Mean and Variance, Standard deviation 35 min
        • Lecture16.3
          Ex-15.2 , Variance and Standard deviation 09 min
        • Lecture16.4
          Ex-15.3, Analysis of frequency distribution, comparison of two frequency distribution with same mean 23 min
        • Lecture16.5
          Chapter Notes – Statistics
      • 17.Probability
        3
        • Lecture17.1
          Outcomes & sample space, Ex. 16.3 19 min
        • Lecture17.2
          Ex.16.3, Probability of an event, Algebra of event 38 min
        • Lecture17.3
          Chapter Notes – Probability
      • 18.Binary Number
        2
        • Lecture18.1
          Binary numbers, Conversion of Binary to Decimal and Decimal to binary 45 min
        • Lecture18.2
          Addition, Subtraction, Multiplication, Division 02 min

        Chapter Notes – Conic Sections Circles

        Circles

        The circle is defined as the locus of a point which moves in a plane such that its distance from a fixed point in that plane is constant.

        Standard Forms of a Circle

        1. Equation of circle having centre (h, k) and radius (x — h)2 + (y — k)2 = a2.

        If centre is (0, 0), then equation of circle is x2 + y2 = a2.

        1. When the circle passes through the origin, then equation of the circle is x2 + y2 — 2hx — 2ky = 0.

        1. When the circle touches the X-axis, the equation is x2 + y2 — 2hx — 2ay + h2 = O.

        1. Equation of the circle, touching the Y-axis is x2 + y2 — 2ax — 2ky + k2 = 0.

        1. Equation of the circle, touching both axes is x2 + y2 — 2ax — 2ay + a2 = O.

        1. Equation of the circle passing through the origin and centre lying on the X-axis is x2 + y2 — 2ax = O.

        1. Equation of the circle passing through the origin and centre lying on the Y-axis is x2 + y2 – 2ay = 0.

        1. Equation of the circle through the origin and cutting intercepts a and b on the coordinate axes is x2 + y2 — by = 0.

         

        1. Equation of the circle, when the coordinates of end points of a diameter are (x1, y1) and (x2, y2) is

        (x — x1)(x — x2) + (y – y1)(y — y2) = 0.

        1. Equation of the circle passes through three given points (x1, y1), (x2, y2) and (x3, y3) is

        1. Parametric equation of a circle (x – h)2 + (y – k)2 = a2 is

        x = h + a cosθ, y = k + a sinθ, 0 ≤ θ ≤ 2π

        For circle x2 + y2 = a2, parametric equation is x = a cos θ, y = a sin θ

        General Equation of a Circle

        The general equation of a circle is given by x2 + y2 + 2gx + 2fy + c = 0, where centre of the circle = (- g, – f)

        Radius of the circle = √g2 + f2 – c

          1. If g2 + f2 – c > 0, then the radius of the circle is real and hence the circle is also real.
          2. If g2 + f2 – c = 0, then the radius of the circle is 0 and the circle is known as point circle.
          3. If g2 + f2 – c< 0, then the radius of the circle is imaginary. Such a circle is imaginary, which is not possible to draw.

        Position of a Point with Respect to a Circle

        A point (x1, y1) lies outside on or inside a circle

        S ≡ x2 + y2 + 2gx + 2fy + c = 0, according as S1 > , = or < 0 where, S1 = x 2 + y 2 + 2gx , + 2fy + c

        Intercepts on the Axes

        The length of the intercepts made by the circle x2 + y2 + 2gx + 2fy + c = 0 with X and Y-axes are

        2√g2 – c and 2√g2 – c.

        1. If g2 > c, then the roots of the equation x2 + 2gx + c = 0 are real and distinct, so the circle x2 + y2 + 2gx + 2fy + c = 0 meets the X-axis in two real and distinct points.
        2. If g2 = c, then the roots of the equation x2 + 2gx + c = 0 are real and equal, so the circle touches X-axis, then intercept on X-axis is O.
        3. If g2 < c, then the roots of the equation x2 + 2gx + c = 0 are imaginary, so the given circle does not meet X-axis in real point. Similarly, the circle x2 + y2 + 2gx + 2fy + c = 0 cuts the Y-axis in real and distinct points touches or does not meet in real point according to f2 >, = or < c

        Equation of Tangent

        A line which touch only one point of a circle.

        Point Form

          1. The equation of the tangent at the point P(x1, y1) to a circle x2 + y2 2gx + 2fy + c= 0 is xx1 + yy1 + g(x + x1) + f(y + y1) + c = 0
          2. The equation of the tangent at the point P(x1, y1) to a circle x2 + y2 is xx1 + yy1 = r2

        Slope Form

        1. The equation of the tangent of slope m to the circle x2 + y2 + 2gx + 2fy + c = 0 are y + f = m(x + g) ± √(g2 + f2 — c)(1 + m2)
        2. The equation of the tangents of slope m to the circle (x – a)2 (y – b)2 = r2 are y – b = m(x –

        a) ± r√(1 + m2) and the coordinates of the points of contact are

        1. The equation of tangents of slope m to the circle x2 + y2 = r2 are y = mx ± r√(1 + m2) and the coordinates of the point of contact are

         

        Parametric Form

        The equation of the tangent to the circle (x – a)2 + (y – b)2 = r2 at the point (a + r cos θ, b + r sinθ) is (x – a) cos θ + (y – b) sin θ = r.

        Equation of Normal

        A line which is perpendicular to the tangent.

        Point Form

          1. (i) The equation of normal at the point (x1, y1) to the circle x2 + y2 + 2gx + 2fy + c = 0 is y – y1 = [(y1 + f)(x – x1)]/(x1 + g)

        (y1 + f)x – (x1 + g)y + (gy1 – fx1) = 0

          1. (ii) The equation of normal at the point (x1, y1) to the circle x2 + y2 = r2 is x/x1 = y/y1

        Parametric Form

        The equation of normal to the circle x2 + y2 = r2 at the point (r cos θ, r sin θ) is (x/r cos θ) = (y/r sin θ)

        or y = x tan θ.

        Important Points to be Remembered

        1. The line y = mx + c meets the circle in unique real point or touch the circle x2 + y2 + r2, if r = |c/√1 + m2

        and the point of contacts are 

        1. The line lx + my + n = 0 touches the circle x2 + y2 = r2, if r2(l2 + m2) = n2.
        2. Tangent at the point P (θ) to the circle x2 + y2 = r2 is x cos θ + y sin θ = r.
        3. The point of intersection of the tangent at the points P(θ1) and Q(θ2) on the circle x2 + y2 = r2

        1. Normal at any point on the circle is a straight line which is perpendicular to the tangent to the curve at the point and it passes through the centre of circle.
        2. Power of a point (x1, y1) with respect to the circle x2 + y2 + 2gx + 2fy + c = 0 is x 2 + y12 +

        1

        2gx1 + 2fy1 + c.

        1. If P is a point and C is the centre of a circle of radius r, then the maximum and minimum distances of P from the circle are CP + r and CP — r , respectively.
        2. If a line is perpendicular to the radius of a circle at its end points on the circle, then the line is a tangent to the circle and vice-versa.

        Pair of Tangents

        1. The combined equation of the pair of tangents drawn from a point P(x1, y1) to the circle x2 + y2 = r2 is

        (x2 + y2 – r2)(x 2+ y 2 – r 2) = (xx + yy

        – r2)2

        1 1 1 1 1

        or SS1 = T2

        where, S = x2 + y2 – r2, S1 = x 2+ y 2 – r 2

        and T = xx1 + yy1 – r2

        1 1 1

        1. The length of the tangents from the point P(x1, y1) to the circle x2 + y2 + 2gx + 2fy + c = 0 is equal to

        1. Chord of contact TT’ of two tangents, drawn from P(x1, y1) to the circle x2 + y2 = r2 or T = 0.

        Similarly, for the circle

        x2 + y2 + 2gx + 2fy + c = 0 is

        xx1 + yy1 + g(x + x1) + f(y + y1) + c = 0

         

        1. Equation of Chord Bisected at a Given Point The equation of chord of the circle S ≡ x2 + y2 + 2gx + 2fy + c = 0 bisected at the point (x1, y1) is give by T = S1.

        i.e., xx1 + yy1 + g (x + x1) + f (y + y1) + c

        = x12 + y12 + 2gx1 + fy1 + c

        1. Director Circle The locus of the point of intersection of two perpendicular tangents to a given circle is called a director circle. For circle x2 + y2 = r2, the equation of director circle is x2 + y2 = 2r2.

        Common Chord

        The chord joining the points of intersection of two given circles is called common chord.

        1. If S1 = 0 and S1 = 0 be two circles, such that

        S1 ≡ x2 + y2 + 2g1x + 2f1y + c1 = 0 and S2 ≡ x2 + y2 + 2g2x + 2f2y + c2 = 0

        then their common chord is given by S1 — S2 = 0

        1. If C1, C2 denote the centre of the given circles, then their common chord PQ = 2 PM = 2√(C1P)2 – C1M)2
        2. If r1, and r2 be the radii of ‘two circles, then length of common chord is

        Angle of Intersection of Two Circles

        The angle of intersection of two circles is defined as the angle between the tangents to the two circles at their point of intersection is given by

        cos θ = (r12 + r22 – d2)/(2r1r2)

        Orthogonal Circles

        Two circles are said to be intersect orthogonally, if their angle of intersection is a right angle. If two circles

        S1 ≡ x2 + y2 + 2g1x + 2f1y + C1 = 0 and

        S2 ≡ x2 + y2 + 2g2x + 2f2y + C2 = 0 are orthogonal, then 2g1g2 + 2f1f2 = c1 + c2

        Family of Circles

        1. The equation of a family of circles passing through the intersection of a circle x2 + y2 + 2gx

        + 2fy + c = 0 and line

        L = lx + my + n = 0 is S + λL = 0 where, X, is any real number.

        1. The equation of the family of circles passing through the point A(x1, y1) and B (x1, y1) is

        1. The equation of the family of -circles touching the circle S ≡ x2 + y2 + 2gx + 2fy + c = 0 at point P(x1, Y1) is

        xx2 + y2 + 2gx + 2fy + c + λ, [xx1 + yy1 + g(x + x1) + f(Y+ Y1) + c] = 0 or S + λL = 0, where L

        = 0 is the equation of the tangent to S = 0 at (x1, y1) and X ∈ R

        1. Any circle passing through the point of intersection of two circles S1 and S2 is S1 +λ(S1 — S2) = 0.

        Radical Axis

        The radical axis of two circles is the locus of a point which moves in such a way that the length of the tangents drawn from it to the two circles are equal.

        A system of circles in which every pair has the same radical axis is called a coaxial system of circles.

        The radical axis of two circles S1 = 0 and S2 = 0 is given by S1 — S2 = 0.

          1. The radical axis of two circles is always perpendicular to the line joining the centres of the circles.
          2. The radical axis of three vertices, whose centres are non-collinear taken in pairs of concurrent.
          3. The centre of the circle cutting two given circles orthogonally, lies on their radical axis.
          4. Radical Centre The point of intersection of radical axis of three circles whose centre are non-collinear, taken in pairs, is called their radical centre.

         

        Pole and Polar

        If through a point P (x1, y1) (within or outside a circle) there be drawn any straight line to meet the given circle at Q and R, the locus of the point of intersection of tangents at Q and R is called the polar of P and po.:.at P is called the pole of polar.

        1. Equation of polar to the circle x2 + y2 = r2 is xx1 + yy1 = r2.
        2. Equation of polar to the circle x2 + y2 + 2gx + 2fy + c = 0 is xx1 + yy1 + g(x + x1) + f(y + y1) + c = 0
        3. Conjugate Points Two points A and B are conjugate points with respect to a given circle, if each lies on the polar of the other with respect to the circle.
        4. Conjugate Lines If two lines be such that the pole of one lies on the other, then they are called conjugate lines with respect to the given circle.

        Coaxial System of Circles

        A system of circle is said to be coaxial system of circles, if every pair of the circles in the system has same radical axis.

        1. The equation of a system of coaxial circles, when the equation of the radical axis P ≡ lx

        + my + n = 0 and one of the circle of the system S = x2 + y2 + 2gx + 2fy + c = 0, is S + λP = 0.

        1. Since, the lines joining the centres of two circles is perpendicular to their radical axis. Therefore, the centres of all circles of a coaxial system lie on a straight line, which is perpendicular to the common radical axis.

        Limiting Points

        Limiting points of a system of coaxial circles are the centres of the point circles belonging to the family.

        Let equation of circle be x2 + y2 + 2gx + c = 0

        ∴ Radius of circle = √g2 — c For limiting point, r = 0

        ∴ √g2 — c = 0 &rArr;g = ± √c

        Thus, limiting points of the given coaxial system as (√c, 0) and (—√c, 0).

        Important Points to be Remembered

        1. Circle touching a line L=O at a point (x1, y1) on it is (x — x1)2 + (y — y1)2 + XL = 0.
        2. Circumcircle of a A with vertices (x1, y1), (x2, y2), (x3, y3) is

        1. A line intersect a given circle at two distinct real points, if the length of the perpendicular from the centre is less than the radius of the circle.
        2. Length of the intercept cut off from the line y = mx + c by the circle x2 + y2 = a2 is

        1. In general, two tangents can be drawn to a circle from a given point in its plane. If m1 and m2 are slope of the tangents drawn from the point P(x1, y1) to the circle x2 + y2 = a2, then

        1. Pole of lx + my + n = 0 with respect to x2 + y2 = a2 is 
        2. Let S1 = 0, S2 = 0 be two circles with radii r1 , r2, then S1/r1 ± S2/r2 = 0 will meet at right angle.
        3. The angle between the two tangents from (x1, y1) to the circle x2 + y2 = a2 is 2 tan– 1 (a/√S1).
        4. The pair of tangents from (0, 0) to the circle x2 + y2 + 2gx + 2fy + c = 0 are at right angle, if g2 + f2 = 2c.
        5. If (x1, y1) is one end of a diameter of the circle x2 + y2 + 2gx + 2fy + c = 0, then the other end will be (-2g – x1, -2f – y1).

         

        Image of the Circle by the Line Minor

        Let the circle be x2 + y2 + 2gx + 2fy + c = 0

        and line minor lx + my + n = 0. Then, the image of the circle is (x — X1)2 + (y — y1)2 =r2

        where, r = √g2 + f2 — c

        Diameter of a Circle

        The locus of the middle points of a system of parallel chords of a circle is called a diameter of the circle.

        1. The equation of the diameter bisecting parallel chords y = mx + c of the circle x2 + y2 = a2 is x + my = 0.
        2. The diameter corresponding to a system of parallel chords of a circle always passes through the centre of the circle and is perpendicular to the parallel chords.

        Common Tangents of Two Circles

        Let the centres and radii of two circles are C1, C2 and r1, r2, respectively.

          1. (i) When one circle contains another circle, no common tangent is possible. Condition, C1C2 < r1 – r2
          2. (ii) When two circles touch internally, one common tangent is possible. Condition , C1C2 = r1 – r2
          3. (iii) When two circles intersect, two common tangents are possible. Condition, |r1 — r2| < C1C2 < |r1 + r2|
          4. (iv) When two circles touch externally, three common tangents are possible. Condition, C1C2 = r1 + r2
          5. (v) When two circles are separately, four common tangents are possible. Condition, C1C2 > r1 + r2

         

        Important Points to be Remembered

        Let AS is a chord of contact of tangents from C to the circle x2 + y2 = r2. M is the mid-point of AB.

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