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1|Page, , Maths Class 11 Chapter 5 Part -1 Quadratic equations, 1. Real Polynomial: Let a0, a1, a2, … , an be real numbers and x is a real variable. Then, f(x) =, a0 + a1x + a2x2 + … + anxn is called a real polynomial of real variable x with real coefficients., 2. Complex Polynomial: If a0, a1, a2, … , an be complex numbers and x is a varying complex, number, then f(x) = a0 + a1x + a2x2 + … + an – 1xn – 1 + anxn is called a complex polynomial or a, polynomial of complex variable with complex coefficients., 3. Degree of a Polynomial: A polynomial f(x) = a0 + a1x + a2x2 + a3x3 + … + anxn , real or, complex is a polynomial of degree n , if an ≠ 0., 4. Polynomial Equation: If f(x) is a polynomial, real or complex, then f(x) = 0 is called a, polynomial equation. If f(x) is a polynomial of second degree, then f(x) = 0 is called a, quadratic equation ., Quadratic Equation: A polynomial of second degree is called a quadratic polynomial., Polynomials of degree three and four are known as cubic and biquadratic polynomials, respectively. A quadratic polynomial f(x) when equated to zero is called quadratic equation., i.e., ax2 + bx + c = 0 where a ≠ 0., Roots of a Quadratic Equation: The values of variable x .which satisfy the quadratic equation, is called roots of quadratic equation., Important Points to be Remembered, , , , , , , , , An equation of degree n has n roots, real or imaginary ., Surd and imaginary roots always occur in pairs of a polynomial equation with real, coefficients i.e., if (√2 + √3i) is a root of an equation, then’ (√2 – √3i) is also its root. ., An odd degree equation has at least one real root whose sign is opposite to that of its, last’ term (constant term), provided that the coefficient of highest degree term is, positive., Every equation of an even degree whose constant term is negative and the coefficient of, highest degree term is positive has at least two real roots, one positive and one negative., If an equation has only one change of sign it has one positive root., If all the terms of an equation are positive and the equation involves odd powers of x,, then all its roots are complex., , Solution of Quadratic Equation, 1.Factorization Method: Let ax2 + bx + c = α(x – α) (x – β) = O. Then, x = α and x = β will, satisfy the given equation., 2. Direct Formula: Quadratic equation ax2 + bx + c = 0 (a ≠ 0) has two roots, given by

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2|Page, , where D = Δ = b2 – 4ac is called discriminant of the equation ., Above formulas also known as Sridharacharya formula., Nature of Roots, Let quadratic equation be ax2 + bx + c = 0, whose discriminant is D., (i) For ax2 + bx + c = 0; a, b , C ∈ R and a ≠ 0, if, (a) D < => Complex roots, (b) D > 0 => Real and distinct roots, (c) D = 0 => Real and equal roots as α = β = – b/2a, (ii) If a, b, C ∈ Q, a ≠ 0, then, (a) If D > 0 and D is a perfect square => Roots are unequal and rational., (b) If D > 0, a = 1; b, c ∈ I and D is a perfect square. => Roots are integral. ., (c) If D > and D is not a perfect square. => Roots are irrational and unequal., (iii) Conjugate Roots The irrational and complex roots of a quadratic equation always occur in, pairs. Therefore,, (a) If one root be α + iβ, then other root will be α – iβ., (b) If one root be α + √β, then other root will be α – √β., (iv) If D, and D2 be the discriminants of two quadratic equations, then, (a) If D1 + D2 ≥ 0, then At least one of D1 and D2 ≥ 0 If D1 < 0, then D2 > 0 ,, (b) If D1 + D2 < 0, then At least one of D1 and D2 < 0 If D1 > 0, then D2 < 0, Roots Under Particular Conditions, For the quadratic equation ax2 + bx + e = 0.

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3|Page, , (i) If b = 0 => Roots are real/complex as (c < 0/c > 0) and equal in magnitude but of opposite, sign., (ii) If c = 0 => One roots is zero, other is – b / a., (iii) If b = C = 0 => Both roots are zero., (iv) If a = c => Roots are reciprocal to each other., (v) If a > 0, c < 0, a < 0, c > 0} => Roots are of opposite sign., (vi) If a > 0, b > 0, c > 0, a < 0, b < 0, c < 0} => Both roots are negative, provided D ≥ 0, (vii) If a > 0, b < 0, c > 0, a < 0, b > 0, c < 0} => Both roots are positive, provided D ≥ 0, (viii) If sign of a = sign of b ≠ sign of c => Greater root in magnitude is negative., (ix) If sign of b = sign of c ≠. sign of a => Greater root in magnitude is positive., (x) If a + b + c = 0 => One root is 1 and second root is c/a., Relation between Roots and Coefficients, 1. Quadratic Equation: If roots of quadratic equation ax2 + bx + c = 0 (a ≠ 0) are α and β, then, Sum of roost = S = α + β = -b/a = – coefficient of x / coefficient of x2 Product of roots = P = α, * β = c/a = constant term / coefficient of x2, 2. Cubic Equation: If α, β and γ are the roots of cubic equation ax3 + bx2 + cx + d = 0., Then,, , 3. Biquadratic Equation: If α, β, γ and δ are the roots of the biquadratic equation ax4 + bx3 +, cx2 + dx + e = 0, then

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4|Page, , Symmetric Roots: If roots of quadratic equation ax2 + bx + c = 0 (a ≠ 0) are α and β, then, , Formation of Polynomial Equation from Given Roots, If a1, a2 a3,…, an are the roots of an nth degree equation, then the equation is xn – S1Xn – 1 +, S2Xn – 2 – S3Xn – 3 +…+( _l)n Sn = 0 where Sn denotes the sum of the products of roots taken n at, a time., 1. Quadratic Equation

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6|Page, , If α be the common root of quadratic equations, a1x2 + b1x + C1 = 0,, and a2x2 + b2x + C2 = 0,, then a1a2 + b1α + C1 = 0,, and a2a2 + b2α + C2 = 0,, By Cramer’s Rule, , Hence, the condition for only one root common is, (c1a2 – c2a1)2 = (b1c2 – b2c1)(a1b2 – a2b1), 2. Both Roots are Common, The required condition is, a1 / a2 = b1 / b2 = c1 / c2, (i) To find the common root of two equations, make the coefficient of second degree term in, the two equations equal and subtract. The value of x obtained is the required common root., (ii) Two different quadratic equations with rational coefficient can not have single common, root which is complex or irrational as imaginary and surd roots always occur in pair., Properties of Quadratic Equation, (i) f(a) . f(b) < 0, then at least one or in general odd number of roots of the equation f(x) = 0 lies, between a and b.

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7|Page, , (ii) f( a) . f( b) > 0, then in general even number of roots of the equation f(x) = 0 lies between a, and b or no root exist f(a) = f(b), then there exists a point c between a and b such that f'(c) = 0,, a < c < b., (iii) If the roots of the quadratic equation a1x2 + b1x + c1 = 0, a2x2 + b2x + c2 = 0 are in the ratio, (i.e., &alpha1;/β1 = &alpha2;/β2), then, b12 / b22 = a1c1 / a2c2., (iv) If one root is k times the other root of the quadratic equation ax2 + bx + c = 0 ,then, (k + 1)2 / k = b2 / ac, Quadratic Expression, An expression of the form ax2 + bx + c, where a, b, c ∈ R and a ≠ 0 is called a quadratic, expression in x ., 1. Graph of a Quadratic Expression, We have, y = ax2 + bx + c = f(x)

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8|Page, , Let y + D/4a = Y and x + D / 2a = X, Y = a * X2 => X2 = Y / a, (i) The graph of the curve y = f(x) is parabolic., (ii) The axis of parabola is X = 0 or x + b / 2a = 0 i.e., (parallel to Y-axis). •, (iii) If a > 0, then the parabola opens upward., If a < 0, then the parabola opens downward., , 2. Position of y = ax2 +bx + c with Respect to Axes., (i) For D > 0, parabola cuts X-axis in two real and distinct points, i.e, x = -b ± √D / 2a, , (ii) For D = 0, parabola touch X-axis in one point, x = – b/2a., , (iii) For D < O,parabola does not cut X-axis (i.e., imaginary value of x).

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9|Page, , 3. Maximum and Minimum Values of Quadratic Expression, (i) If a > 0, quadratic expression has least value at x = b / 2a. This least value is given by 4ac –, b2 / 4a = – D/4a. But their is no greatest, value., (ii) If a < 0, quadratic expression has greatest value at x = – b/2a . This greatest value is given, by 4ac – b2 / 4a = – D/4a. But their is no least value., 4. Sign of Quadratic Expression, (i) a > 0 and D < 0, so f(x) > 0 for all x ∈ R i.e., f(x) is positive for all real values of x., (ii) a < 0 and D < 0, so f(x) < 0 for all x ∈ R i.e., f(x) is negative for all real values of x., (iii) a > 0 and D = 0, so f(x) ≥ 0 for all x ∈ R i.e., f(x) is positive for all real values of x except, at vertex, where f(x) = 0., (iv) a < 0 and D = 0, so f(x) ≤ 0 for all x ∈ R i.e., f(x) is negative for all real values of x except, at vertex, where f(x) = 0., (v) a > 0 and D > 0, Let f(x) = o have two real roots α and β (α < β), then f(x) > 0 for x ∈ (- ∞, α) ∪ (β,∞) and f (x), < 0 for all x ∈ (α, β)., (vi) a < 0 and D > 0, Let f(x) = 0 have two real roots α and β (α < β). Then, f(x) < 0 for all x ∈ (- ∞, α) ∪ (β,∞) and, f(x) > 0 for all, x ∈ (α, β). ,, 5. Intervals of Roots, In some problems, we want the roots of the equation ax2 + bx + c = 0 to lie in a given interval., For this we impose conditions on a, b and c., Since, a ≠ 0, we can take f(x) = x2 + b/a x + c/a., (i) Both the roots are positive i.e., they lie in (0,∞), if and only if roots are real, the sum of the, roots as well as the product of the roots is positive., α + β = -b/a > 0 and αβ = c/a > 0 with b2 – 4ac ≥ 0, Similarly, both the roots are negative i.e., they lie in (- ∞,0) ifF roots are real, the sum of the, roots is negative and the product of the roots is positive., i.e., α + β = -b/a < 0 and αβ = c/a > 0 with b2 – 4ac ≥ 0

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10 | P a g e, , (ii) Both the roots are greater than a given number k, iFf the following conditions are satisfied, D ≥ 0, -b/2a > k and f(k) > 0, , (iii) Both .the roots are less than a given number k, iff the following conditions are satisfied, D ≥ 0, -b/2a > k and f(k) > 0, (iv) Both the roots lie in a’ given interval (k1, k2), iff the following conditions are satisfied, D ≥ 0,k1 < -b/2a < k2 and f(k1) > 0, f(k2) > 0, , (v) Exactly one of the roots lie in a given interval (k1, k2), iff, f(k1) f(k2) < 0, , (vi) A given number k lies between the roots iff f(k) < O. In particular, the roots of the equation, will be of opposite sign, iff 0 lies between the roots., ⇒ f(0) < 0, , Wavy Curve Method, Let f(x) = (x – a1)k1 (x – a2)k2(x — a3)k3 … (x – an – 1)kn – 1 (x – an)kn

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11 | P a g e, , where k1, k2, k3,…, kn ∈ N and a1, a2, a3,…, an are fixed natural numbers satisfying the, condition., a1 < a2 < a3 < … < an – 1 < an.., First we mark the numbers a1, a2, a3,…, an on the real axis and the plus sign in the interval of, the right of the largest of these numbers, i.e., on the right of an. If kn is even, we put plus sign, on the left of anand if kn is odd, then we put minus sign on the left of an In the next interval we, put a sign according to the following rule., When passing through the point an – 1 the polynomial f(x) changes sign . if kn – 1 is an odd, number and the polynomial f(x) has same sign if kn – 1 is an even number. Then, we consider, the next interval and put a sign in it using the same rule., Thus, we consider all the intervals. The solution of f(x) > 0 is the union of all interval in which, we have put the plus sign and the solution of f(x) < 0 is the union of all intervals in which we, have put the minus Sign., Descarte’s Rule of Signs, The maximum number of positive real roots of a polynomial equation f(x) = 0 is the number of, changes of sign from positive to negative and negative to positive in f(x) ., The maximum number of negative real roots of a polynomial equation f(x) = 0 is the number of, changes of sign from positive to negative and negative to positive in f(x)., Rational Algebraic In equations, (i) Values of Rational Expression P(x)/Q(x) for Real Values of x, where P(x) and Q(x) are, Quadratic Expressions To find the values attained by rational expression of the form a1x2 +, b1x + c1 / a2x2 + b2x + c2, for real values of x., (a) Equate the given rational expression to y., (b) Obtain a quadratic equation in x by simplifying the expression,, (c) Obtain the discriminant of the quadratic equation., (d) Put discriminant ≥ 0 and solve the in equation for y. The values of y so obtained determines, the set of values attained by the given rational expression., (ii) Solution of Rational Algebraic In equation If P(x) and Q(x) are polynomial in x, then the, in equation P(x) / Q(x) > 0,, P(x) / Q(x) < 0, P(x) / Q(x) ≥ 0 and P(x) / Q(x) ≤ 0 are known as rational algebraic in, equations., To solve these in equations we use the sign method as

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12 | P a g e, , (a) Obtain P(x) and Q(x)., (b) Factorize P(x) and Q(x) into linear factors., (c) Make the coefficient of x positive in all factors., (d) Obtain critical points by equating all factors to zero., (e) Plot the critical points on the number line. If these are n critical points, they divide the, number line into (n + 1) regions., (f) In the right most region the expression P(x) / Q(x) bears positive sign and in other region the, expression bears positive and negative signs depending on the exponents of the factors ., Lagrange’s identity, If a1, a2, a3, b1, b2, b3 ≠ R, then, (a12 + a22 + a32) (b12 + b22 + b32) – (a1b1 + a2b2 + a3b3)2, = (a1b2 – a2b1)2 + (a2b3 – a3b2 )2 + (a3b1 – a1b3)2, Algebraic Interpretation of Rolle’s Theorem, Let f (x) be a polynomial having α and β as its roots such that α < β, f(α) = f(β) = 0.Also, a, polynomial function is everywhere continuous and differentiable, then there exist θ ∈ (α, β), such that f'(θ) = 0. Algebraically, we can say between any two zeros of a polynomial f(x) there, is always a derivative f’ (x) = 0., Equation and In equation Containing Absolute Value, 1. Equation Containing Absolute Value, By definition, |x| = x, if x ≥ 0 OR -x, if x < 0, If |f(x) + g(x)| = |f(x)| + g(x)|, then it is equivalent to the system f(x) . g(x) ≥ 0., If |f(x) – g(x)| = |f(x)| – g(x)|, then it is equivalent to the system f(x) . g(x) ≤ 0., 2.In equation Containing Absolute Value, (i) |x| < a ⇒ – a < x < a (a > 0), (ii) |x| ≤ a ⇒ – a ≤ x ≤ a, (iii) |x| > a ⇒ x < – a or x > a, (iv) |x| ≥ a ⇒ x le; – a or x ≥ a, 3. Absolute Value of Real Number, |x| = -x, x < 0 OR +x, x ≥ 0

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13 | P a g e, , (i) |xy| = |x||y|, (ii) |x / y| = |x| / |y|, (iii) |x|2 = x2, (iv) |x| ≥ x, (v) |x + y| ≤ |x| + |y|, Equality hold when x and y same sign., (vi) |x – y| ≥ ||x| – |y||, Inequalities, Let a and b be real numbers. If a – b is negative, we say that a is less than b (a < b) and if a – b, is positive, then a is greater than b (a > b)., Important Points to be Remembered, (i) If a > b and b > c, then a > c. Generally, if a1 > a2, a2 > a3,…., an – 1 > an, then a1 > an., , (vii) If a < x < b and a, b are positive real numbers then a2 < x2 < b2

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14 | P a g e, , Important Inequality, 1. Arithmetico-Geometric and Harmonic Mean Inequality, (i) If a, b > 0 and a ≠ b, then, , (ii) if ai > 0, where i = 1,2,3,…,n, then, , (iii) If a1, a2,…, an are n positive real numbers and m1, m2,…,mn are n positive rational, numbers, then

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1|Page, , Maths Class 11 Chapter 5 Part -1 Complex Numbers, Imaginary Quantity, The square root of a negative real number is called an imaginary quantity or imaginary number., e.g., √-3, √-7/2, The quantity √-1 is an imaginary number, denoted by ‘i’, called iota., Integral Powers of Iota (i), i=√-1, i2 = -1, i3 = -i, i4=1, So, i4n+1= i, i4n+2 = -1, i4n+3 = -i, i4n+4 = i4n = 1, In other words,, in = (-1)n/2, if n is an even integer, in = (-1)(n-1)/2.i, if is an odd integer, Complex Number, A number of the form z = x + iy, where x, y ∈ R, is called a complex number, The numbers x and y are called respectively real and imaginary parts of complex number z., i.e., x = Re (z) and y = Im (z), Purely Real and Purely Imaginary Complex Number, A complex number z is a purely real if its imaginary part is 0., i.e., Im (z) = 0. And purely imaginary if its real part is 0 i.e., Re (z)= 0., Equality of Complex Numbers, Two complex numbers z1 = a1 + ib1 and z2 = a2 + ib2 are equal, if a2= a2 and b1 = b2 i.e., Re (z1), = Re (z2) and Im (z1) = Im (z2)., Algebra of Complex Numbers, 1. Addition of Complex Numbers, Let z1 = (x1 + iyi) and z2 = (x2 + iy2) be any two complex numbers, then their sum defined as, z1 + z2 = (x1 + iy1) + (x2 + iy2) = (x1 + x2) + i(y1 + y2)

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3|Page, , where z2 # 0., Conjugate of a Complex Number, If z = x + iy is a complex number, then conjugate of z is denoted by z, i.e., z = x – iy, Properties of Conjugate, , Modulus of a Complex Number, If z = x + iy, , then modulus or magnitude of z is denoted by |z| and is given by, |z| = x2 + y2.

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4|Page, , It represents a distance of z from origin., In the set of complex number C, the order relation is not defined i.e., z1> z2 or zi <z2 has no, meaning but |z1|>|z2| or |z1|< | z2 | has got its meaning, since |z| and |z2| are real numbers., Properties of Modulus

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5|Page, , Reciprocal/Multiplicative Inverse of a Complex Number, Let z = x + iy be a non-zero complex number, then, , Here, z-1 is called multiplicative inverse of z., Argument of a Complex Number, Any complex number z=x+iy can be represented geometrically by a point (x, y) in a plane,, called Argand plane or Gaussian plane. The angle made by the line joining point z to the origin,, with the x-axis is called argument of that complex number. It is denoted by the symbol arg (z), or amp (z)., , Argument (z) = θ = tan-1(y/x), Argument of z is not unique, general value of the argument of z is 2nπ + θ. But arg (0) is not, defined., A purely real number is represented by a point on x-axis., A purely imaginary number is represented by a point on y-axis., There exists a one-one correspondence between the points of the plane and the members of the, set C of all complex numbers., The length of the line segment OP is called the modulus of z and is denoted by |z|., i.e., length of OP = √x2 + y2., Principal Value of Argument

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6|Page, , The value of the argument which lies in the interval (- π, π] is called principal value of, argument., (i) If x> 0 and y > 0, then arg (z) = 0, (ii) If x < 0 and y> 0, then arg (z) = π -0, (iii) If x < 0 and y < 0, then arg (z) = – (π – θ), (iv) If x> 0 and y < 0, then arg (z) = -θ, Properties of Argument

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7|Page

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8|Page, , Square Root of a Complex Number, If z = x + iy, then, , Polar Form, If z = x + iy is a complex number, then z can be written as, z = |z| (cos θ + i sin θ) where, θ = arg (z), this is called polar form., If the general value of the argument is 0, then the polar form of z is, z = |z| [cos (2nπ + θ) + i sin (2nπ + θ)], where n is an integer., Eulerian Form of a Complex Number, If z = x + iy is a complex number, then it can be written as, z = rei0, where, r = |z| and θ = arg (z), This is called Eulerian form and ei0= cosθ + i sinθ and e-i0 = cosθ — i sinθ., De-Moivre’s Theorem, A simplest formula for calculating powers of complex number known as De-Moivre’s theorem., If n ∈ I (set of integers), then (cosθ + i sinθ)n = cos nθ + i sin nθ and if n ∈ Q (set of rational, numbers), then cos nθ + i sin nθ is one of the values of (cos θ + i sin θ)n.

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9|Page, , The nth Roots of Unity, The nth roots of unity, it means any complex number z, which satisfies the equation zn = 1 or z, = (1)1/n, or z = cos(2kπ/n) + isin(2kπ/n) , where k = 0, 1, 2, … , (n — 1), Properties of nth Roots of Unity, 1., 2., 3., 4., 5., 6., 7., , nth roots of unity form a GP with common ratio e(i2π/n) ., Sum of nth roots of unity is always 0., Sum of nth powers of nth roots of unity is zero, if p is a multiple of n, Sum of pth powers of nth roots of unity is zero, if p is not a multiple of n., Sum of pth powers of nth roots of unity is n, ifp is a multiple of n., Product of nth roots of unity is (-1)(n – 1)., The nth roots of unity lie on the unit circle |z| = 1 and divide its circumference into n, equal parts., , The Cube Roots of Unity, Cube roots of unity are 1, ω, ω2,, where ω = -1/2 + i√3/2 = e(i2π/3) and ω2 = (-1 – i√3)/2, ω3r + 1 = ω, ω3r + 2 = ω2

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10 | P a g e, , Properties of Cube Roots of Unity, (i) 1 + ω + ω2r =, 0, if r is not a multiple of 3., 3, if r is,a multiple of 3., (ii) ω3 = ω3r = 1, (iii) ω3r + 1 = ω, ω3r + 2 = ω2, (iv) Cube roots of unity lie on the unit circle |z| = 1 and divide its circumference into 3 equal, parts., (v) It always forms an equilateral triangle., (vi) Cube roots of – 1 are -1, – ω, – ω2., , Geometrical Representations of Complex Numbers, 1. Geometrical Representation of Addition, If two points P and Q represent complex numbers z1 and z2 respectively, in the Argand plane,, then the sum z1 + z2 is represented, by the extremity R of the diagonal OR of parallelogram OPRQ having OP and OQ as two, adjacent sides.

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12 | P a g e, , R has the polar coordinates (r1/r2, θ1 – θ2) and it represents the complex number z1/z2., |z|=|z| and arg (z) = – arg (z). The general value of arg (z) is 2nπ – arg (z)., If a point P represents a complex number z, then its conjugate i is represented by the image of P, in the real axis., , Concept of Rotation, Let z1, z2 and z3 be the vertices of a ΔABC described in anti-clockwise sense. Draw OP and, OQ parallel and equal to AB and AC, respectively. Then, point P is z2 – z1 and Q is z3 – z1. If, OP is rotated through angle a in anti-clockwise, sense it coincides with OQ., , Important Points to be Remembered

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14 | P a g e, , where z1 = x1 + iy1 and z2 = x2 + iy2, (ii) The point P (z) which divides the join of segment AB in the ratio m : n is given by, z = (mz2 + nz1)/(m + n), If P divides the line externally in the ratio m : n, then, z = (mz2 – nz1)/(m – n), Triangle in Complex Plane, (i) Let ABC be a triangle with vertices A (z1), B(z2) and C(z3 ) then, (a) Centroid of the ΔABC is given by, z = 1/3(z1 + z2 + z3), (b) Incentre of the AABC is given by, z = (az1 + bz2 + cz3)/(a + b + c), (ii) Area of the triangle with vertices A(z1), B(z2) and C(z3) is given by, , For an equilateral triangle,, z12 + z22 + z32 = z2z3 + z3z1 + z1z2, (iii) The triangle whose vertices are the points represented by complex numbers z 1, z2 and z3 is, equilateral, if, , Straight Line in Complex Plane, (i) The general equation of a straight line is az + az + b = 0, where a is a complex number and b, is a real number.

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17 | P a g e, , (iv) If z1, and z2 are the fixed complex numbers, then the locus of a point z satisfying arg [(z –, z1)/(z – z2)] = ± π / 2 is a circle having z1 and z2 at the end points of a diameter., Conic in Complex plane, (i) Let z1 and z2 be two fixed points, and k be a positive real number., If k >|z1- z2|, then |z – z1| + |z – z2| = k represents an ellipse with foci at A(z1) and B(z2) and, length of the major axis is k., (ii) Let z1 and z2 be two fixed points and k be a positive real number., If k ≠ |z1- z2| , then |z – z1| – |z – z2| = k represents hyperbola with foci at A(z1) and B(z2)., Important Points to be Remembered, , , √-a x √-b ≠ √ab, , √a x √b = √ab is possible only, if both a and b are non-negative., So, i2 = √-1 x √-1 ≠ √1, , , , , , , , , , , is neither positive, zero nor negative., Argument of 0 is not defined., Argument of purely imaginary number is π/2, Argument of purely real number is 0 or π., If |z + 1/z| = a then the greatest value of |z| = a + √a2 + 4/2 and the least value of |z| = -a, + √a2 + 4/2, The value of ii = e-π2, The complex number do not possess the property of order, i.e., x + iy < (or) > c + id is, not defined., The area of the triangle on the Argand plane formed by the complex numbers z, iz and z, + iz is 1/2|z|2., (x) If ω1 and ω2 are the complex slope of two lines on the Argand plane, then the lines, are, , (a) perpendicular, if ω1 + ω2 = 0., (b) parallel, if ω1 = ω2.