Important Properties of Conjugate, Modulus & Argument

# Important Properties of Conjugate, Modulus & Argument | Mathematics (Maths) for JEE Main & Advanced PDF Download

D. Important Properties Of Conjugate / Modulus / Argument

If z , z1 , z2C then ;
(a)

(b)

|z| ≥ 0  ;  |z| ≥  Re (z)  ;    |z| ≥ Im (z) ;

(c)

(i) amp (z1 . z2) =  amp  z1 + amp z2 + 2 kπ  . k ∈I

(ii) amp  = amp z1 - amp z2 + 2 kπ   ;     k∈ I\

(iii) amp(zn) = 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 z1 , z2 be two complex numbers represented by points on the circle  respectively , then

Sol.

Maximum value of  are collinear.

Ex.6 Prove that if z1 and z2 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 z2 - z3 = p ; z3 - z1 = q ; z1 - z2 = r ⇒ p + q + r = 0

Given condition, pq + qr + rp = 0   ⇒ p (q + r) + qr = 0   ⇒  p (- p) + qr = 0

⇒ p2 = qr

Ex.8 If z & w are two complex numbers simultaneously satisfying the equations,  then

Sol.

z3 = - w5 ⇒ |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+ y2) (x2 - y2) = 350 ⇒ (x2 - y2) (x2 + y2) = 175 = 35.5 = 25.7

= x2 + y2 = 25 & x2 - y2 = 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 2|z| – 4 az + 1 + ia = 0.

Sol. We have 2|z| – 4 az + 1 + ia = 0
Put z = x + i y,

We get,   = 4 ax – 1 + 4aiy – ia  or 4(x2 + y2) = (4 ax – 1)2    .....(1)

and a = 4 ay (by separating imaginary and real parts)

⇒ y =1/2 and 4x+ 1/4– 16 a2 x2 – 1 + 8 ax = 0  ⇒ x2 (16 - 64 a2) + 32 ax - 3 = 0

The document Important Properties of Conjugate, Modulus & Argument | Mathematics (Maths) for JEE Main & Advanced is a part of the JEE Course Mathematics (Maths) for JEE Main & Advanced.
All you need of JEE at this link: JEE

## Mathematics (Maths) for JEE Main & Advanced

209 videos|443 docs|143 tests

## FAQs on Important Properties of Conjugate, Modulus & Argument - Mathematics (Maths) for JEE Main & Advanced

 1. What is the definition of the conjugate of a complex number?
Ans. The conjugate of a complex number is obtained by changing the sign of its imaginary part. For example, the conjugate of the complex number a + bi is a - bi.
 2. How do you find the modulus of a complex number?
Ans. The modulus (or absolute value) of a complex number a + bi is found by taking the square root of the sum of the squares of its real and imaginary parts. In other words, the modulus of a complex number is given by |a + bi| = √(a^2 + b^2).
 3. What does the argument of a complex number represent?
Ans. The argument of a complex number represents the angle that the complex number makes with the positive real axis in the complex plane. It is usually measured in radians or degrees.
 4. How can you find the argument of a complex number?
Ans. To find the argument of a complex number a + bi, you can use the formula arg(z) = arctan(b/a). However, it is important to consider the quadrant in which the complex number lies in order to obtain the correct argument.
 5. What are some important properties of the conjugate, modulus, and argument of complex numbers?
Ans. Some important properties include: - The conjugate of the conjugate of a complex number is the original complex number. - The product of a complex number and its conjugate is equal to the square of its modulus. - The argument of the product of two complex numbers is equal to the sum of their arguments. - The argument of the quotient of two complex numbers is equal to the difference of their arguments. - The modulus of the product of two complex numbers is equal to the product of their moduli.

## Mathematics (Maths) for JEE Main & Advanced

209 videos|443 docs|143 tests

### Up next

 Explore Courses for JEE exam
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
10M+ students study on EduRev
Related Searches

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

;