D. Important Properties Of Conjugate / Modulus / Argument
If z , z_{1} , z_{2}C then ;
(a)
(b)
z ≥ 0 ; z ≥ Re (z) ; z ≥ Im (z) ;
(c)
(i) amp (z_{1} . z_{2}) = amp z_{1} + amp z_{2} + 2 kπ . k ∈I
(ii) amp = amp z_{1}  amp z_{2} + 2 kπ ; k∈ I\
(iii) amp(z^{n}) = n amp(z) + 2kπ .
where proper value of k must be chosen so that RHS lies in ( π , π ].
Ex.4 The maximum & minimum values of are
Sol. denotes set of points on or inside a circle with centre ( 3, 0) and radius 3.
denotes the distance of P from A
Ex.5 Let z_{1} , z_{2} be two complex numbers represented by points on the circle respectively , then
Sol.
Maximum value of are collinear.
Ex.6 Prove that if z_{1} and z_{2} are two complex numbers and c > 0, then
Sol.
Incorporating the number c > 0, the last term on the RHS can be written
Ex.7 If are the points A, B, C in the Argand Plane such that,
prove that ABC is an equilateral triangle .
Sol . Let z_{2}  z_{3} = p ; z_{3}  z1 = q ; z_{1}  z_{2} = r ⇒ p + q + r = 0
Given condition, pq + qr + rp = 0 ⇒ p (q + r) + qr = 0 ⇒ p ( p) + qr = 0
⇒ p^{2} = qr
Ex.8 If z & w are two complex numbers simultaneously satisfying the equations, then
Sol.
z^{3} =  w^{5} ⇒ z ^{3} =w^{5} ⇒ z ^{6} = w^{10} .....(1)
Ex.9 The complex numbers whose real and imaginary parts are integers and satisfy the relation forms a rectangle on the Argand plane, the length of whose diagonal is
Sol.
= 2 (x^{2 }+ y^{2}) (x^{2}  y^{2}) = 350 ⇒ (x^{2}  y^{2}) (x^{2} + y^{2}) = 175 = 35.5 = 25.7
= x^{2} + y^{2} = 25 & x^{2}  y^{2} = 7 ⇒ x = ± 4 & y = ± 3
Ex.10 Find the area bounded by the curve, arg z in the complex plane .
Sol.
required area, the equilateral triangle OPQ with side 4
Ex.11
Find the complex number where the curves arg intersect.
Sol.
Ex.12 if then prove that is purely real.
Sol. The given relation can be written as
Ex.13 For every real number a ≥ 0 find all the complex numbers z that satisfy the equation 2z – 4 az + 1 + ia = 0.
Sol. We have 2z – 4 az + 1 + ia = 0
Put z = x + i y,
We get, = 4 ax – 1 + 4aiy – ia or 4(x^{2} + y^{2}) = (4 ax – 1)^{2} .....(1)
and a = 4 ay (by separating imaginary and real parts)
⇒ y =1/2 and 4x^{2 }+ 1/4– 16 a^{2} x^{2} – 1 + 8 ax = 0 ⇒ x^{2} (16  64 a^{2}) + 32 ax  3 = 0
209 videos443 docs143 tests

1. What is the definition of the conjugate of a complex number? 
2. How do you find the modulus of a complex number? 
3. What does the argument of a complex number represent? 
4. How can you find the argument of a complex number? 
5. What are some important properties of the conjugate, modulus, and argument of complex numbers? 
209 videos443 docs143 tests


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