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MATHSALGEBRAMathematics FundamentalsComplex Numbers :|z| ≥ |Re(z)| ≥ Re(z) and |z| ≥ |1m(z) | ≥ 1m(z)z| z |is always a unimodular complex number if z ≠ 0|Re(z) | + |1m(z) | ≤If2 |z|1z + = a, the greatest and least values of |z| arezrespectively|z 1 +2 21 z2a +a22 +z − | + |z 2 –4and2 21 z2− a +a22 +z − | = |z 1 + z 2 | + |z 1 – z 2 |If z 1 = z 2 ⇔ |z 1 | = |z 2 | or arg z 1 = arg z 2|z 1 + z 2 | = |z 1 – z 2 | ⇔ arg (z 1 ) – arg(z 2 ) = π/2.If |z 1 | ≤ 1, |z 2 | ≤ 1 then(i) |z 1 + z 2 | 2 ≤ (|z 1 | – |z 2 |) 2 + (arg (z 1 ) – arg (z 2 )) 2(ii) |z 1 + z 2 | 2 ≥ (|z 1 | + |z 2 |) 2 – (arg (z 1 ) – arg(z 2 )) 2|z 1 + z 2 | 2 = |z 1 | 2 + |z 2 | 2 + 2||z 2 | cos(θ 1 – θ 2 ).|z 1 – z 2 | 2 = |z 1 | 2 + |z 2 | 2 – 2|z 1 ||z 2 | cos(θ 1 – θ 2 ).If z 1 and z 2 are two complex numbers then|z 1 z 2 | = r 1 r 2 ; arg(z 1 z 2 ) = θ 1 + θ 2 and⎛ z ⎞arg⎜1⎟ = θ 1 – θ 2 and where |z 1 | = r 1 , |z 2 | = r 2 ,⎝ z2⎠arg(z 1 ) = θ 1 and arg(z 2 ) = θ 2 .The area of the triangle whose vertices are z, iz andz + iz is 21 |z| 2 .The area of the triangle with vertices z, wz and3z + wz is |z 2 |.4If z 1 , z 2 , z 3 be the vertices of an equilateral triangleand z 0 be the circumcentre, then2z 1 +2z 2 +2z 3 = 3 z 2 0 .If z 1 , z 2 , z 3 ....... z n be the vertices of a regularpolygon of n sides and z 0 be its centroid, then2z 1 +2z 2 + .......+2z n = n z 02 .If z 1 , z 2 , z 3 be the vertices of a triangle, then thetriangle is equilateral ifzz12=4rr12,or(z 1 – z 2 ) 2 + (z 2 – z 3 ) 2 + (z 3 – z 1 ) 2 = 02z 1 +2z 2 +2z 3 = z 1 z 2 + z 2 z 3 + z 3 z 11 1 1or + + = 0z 1 − z 2 z 2 − z 3 z 3 − z 1If z 1 , z 2 z 3 are the vertices of an isosceles triangle,2 2 2right angled at z 2 then z 1 + z 2 + z 3 = 2z 2 (z 1 + z 2 )If z 1 , z 2 , z 3 are the vertices of a right-angled isoscelestriangle, then (z 1 – z 2 ) 2 = 2(z 1 – z 3 )(z 3 – z 2 ).If z 1 , z 2 , z 3 be the affixes of the vertices A, B, Crespectively of a triangle ABC, then its orthocentre isa 3(secA)z1+ b(secB)z 2 + (csecC)za sec A + bsecB + csecCFor any a, b ∈ R(i)(ii)a + ib + a − ib = 2{ a + b + a}a + ib – a − ib = 2{ a + b − a}The sum and product of two complex numbers arereal simultaneously if and only if they are conjugateto each other.If ω and ω 2 are the complex cube roots of unity, then(i) (aω + bω 2 )(aω 2 + bω) = a 2 + b 2 – ab(ii) (a + b (aω + bω 2 )(aω 2 + b 2 ω) = a 3 + b 3(iii) (a + bω + cω 2 )(a + bω 2 + cω)= a 2 + b 2 + c 2 – ab – bc – ca(iv) (a + b + c) (a + bω + cω 2 ) (a + bω 2 + cω)= a 3 + b 3 + c 3 – 3abcIf three points z 1 , z 2 , z 3 connected by relationaz 1 + bz 2 + cz 3 = 0 where a + b + c = 0, then the threepoints are collinear.If three complex numbers are in A.P. then they lie ona straight line in the complex plane.Progression :If T k and T p of any A.P. are given, then formula forTn− T Tk p − Tkobtaining T n is = .n − k p − kIf pT p = qT q of an A.P., then T p + q = 0.If p th term of an A.P. is q and the q th term is p, thenT p+q = 0 and T n = p + q – n.If the p th term of an A.P. is 1/q and the q th term is 1/p,then its pq th term is 1.2222XtraEdge for IIT-JEE 48 APRIL 2010