May 2011 - Career Point

−1+i 3If ω = , then the cube roots of unity are 1,2ω and ω 2 . We have:(i) 1 + ω + ω 2 = 0 (ii) ω 3 = 1Let z = x + iy be any complex number.Let z = r (cos θ + i sin θ) where r > 0.∴ x = r cos θ and y = r sin θ∴ x 2 + y 2 = r 2⇒ r =22x + y (Q r > 0)∴ cos θ =x2 2x + yand sin θ =y2 2x + yThe value of θ is found by solving these equations. θis called the argument (or amplitude) of z.If – p < θ ≤ π, then θ is called the principal argumentof z.Identification of θ –x y arg(z) Interval of θ⎛ π ⎞+ + θ ⎜0< θ < ⎟⎝ 2 ⎠⎛ – π ⎞+ – –θ ⎜ < θ < 0⎟ ⎝ 2 ⎠⎛ π ⎞– + (π – θ) ⎜ < θ < π⎟ ⎝ 2 ⎠⎛ – π ⎞– – –(π – θ) ⎜ – π < θ < ⎟⎝ 2 ⎠If z 1 and z 2 are two complex numbers then(i) | z 1 – z 2 | is the distance between the points withaffixes z 1 and z 2 .mz(ii) 2 + nz 1 is the affix of the point dividing them + nline joining the points with affixes z 1 and z 2 in theratio m : n internally.mz2 – nz1(iii)is the affix of the point dividing them – nline joining the points with affixes z 1 and z 2 in theratio m : n externally where m ≠ n.(iv) If z 1 , z 2 , z 3 are the affixes of the vertices of az 1 + z2+ z3triangle then the affix of its centroid is.3(v) z = tz 1 + (1 – t)z 2 is the equation of the line joiningpoints with affixes z 1 and z 2 . Here ‘t’ is a parameter.z − z1z − z1(vi) = is the equation of the linez2− z1z2− z1joining points with affixes z 1 and z 2 .Three points with affixes z 1 , z 2 , z 3 are collinear ifzzz123zzz123111= 0.The general equation of a straight line isa z + az + b = 0 , where b is any real number.(i) | z – z 1 | < r represents the circle with centre z 1and radius r.(ii) | z – z 1 | < r represents the interior of the circlewith centre z 1 and radius r.z − z1= k represents a circle line which is thez − z1perpendicular bisector of the line segment joiningpoints with affixes z 1 and z 2 .(z – z 1 ) ( z − z2)+ ( z − z1)+ (z – z 2 ) = 0 represents thecircle with line joining points with affixes z 1 and z 2 asa diameter.| z – z 1 | + | z – z 2 | = 2k, k ∈ R + represents the ellipsewith foci at points with affixes z 1 and z 2 .If z 1 , z 2 , z 3 be the affixes of the points A, B, Crespectively, then the angle between AB and AC is⎛ z ⎞given by arg⎜3 − z1⎟ .⎝ z2− z1⎠If z 1 , z 2 , z 3 , z 4 are the affixes of the points A, B, C, Drespectively, then the angle between AB and CD is⎛ z ⎞given by arg⎜2 − z1⎟ .⎝ z4− z3⎠nth roots of a complex numberLet z = r (cos θ + i sin θ), r > 0 be any complexnumber. n th root o z = z 1/n= r 1/n ⎛ 2 π + θ kπ + θ ⎞⎜cos k + isin2 ⎟ ,⎝ nn ⎠where k = 0, 1, 2, ………, n – 1.There are n distinct values and sum of all thesevalues is 0.Logarithm of a complex numberLet z = re iθ be any complex number.Then log z = log re iθ = log r + log e iθ= log r + iθ log e = log r + iθ.∴ log z = log | z | + i amp (z).XtraEdge for IIT-JEE 43 MAY 2011