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E. Representation Of A Complex Number
(a) Cartesian Form (Geometric Representation): Every complex number z = x + i y can be represented by a point on the cartesian plane known as complex plane (Argand diagram) by the ordered pair (x , y) .
Length OP is called modulus of the complex number denoted by is called the argument or amplitude.
eg. (angle made by OP with positive xaxis)
Remark :
(i) is always non negative . Unlike real numbers = is not correct
(ii) Argument of a complex number is a many valued function . is the argument of a complex number then will also be the argument of that complex number. Any two arguments of a complex number differ by
(iii) The unique value of θ such that is called the principal value of the argument.
(iv) Unless otherwise stated, amp z implies principal value of the argument.
(v) By specifying the modulus & argument a complex number is defined completely. For the complex number 0 + 0 i the argument is not defined and this is the only complex number which is given by its modulus.
(vi) There exists a oneone correspondence between the points of the plane and the members of the set of complex numbers.
(b) Trigonometric / Polar Representation :
z = r (cos θ+ i sin θ) where z = r ; arg z = θ ; z^{} = r (cos θ  i sin θ)
Remark : cos θ + i sin θ is also written as CiS θ .
are known as Euler's identities.
Ex.14 Express in polar form and then find the modulus and argument of z. Hence deduce the value of
Sol.
Let –1 + i √3 = r(cosθ + i sinθ). Equating real and imaginary parts, r cosθ = 1, r sinθ= √3 .
Now r^{2} = 1 + 3 = 4, r = 2, cosθ = 1/2, sinθ √3/2, or θ = 2π/3 between –π and π.
Consequently, –1 + i √3 =
It is the polar form of z. Obviously, z = √2 and arg z = 5/12 (principal value).
(c) Exponential Representation :
z = re^{iθ } ; z = r ; arg z = θ ; z^{} = re^{ iθ}
(d) Vectorial Representation :Every complex number can be considered as if it is the position vector of that point. If the point P represents the complex number z then,
Ex.15 If (1 – i) is a root of the equation , z^{3}  2 (2  i) z^{2} + (4  5 i) z  1 + 3 i = 0 then find the other two roots.
Sol.
z_{1} + z_{2} + z_{3} = 2 (2 – i) ⇒ z_{2 }+ z_{3 }= 3  i ( z_{1} = 1  i) .......(1)
again z_{1} z_{2} z_{3} = 1 – 3 i
......(2)
From (1) & (2) z_{2} = 1 & z_{3} = 2 – i
Ex.16 Prove that if the ratio is purely imaginary then the point z lies on the circle whose centre is at the point and radius is
Sol. Let z = x + iy.
is purely imaginary, x(x – 1) + y(y – 1) = 0 or
It is a circle with radius and centre
Therefore, the point z lies on a circle and the centre is
Ex.17 A function f is defined on the complex number by f (z) = (a + bi)z, where 'a' and 'b' are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. Given that  a + bi  = 8 and that where u and v are coprimes. Find the value of (u + v).
Sol. Given  (a + bi)z – z  =  (a + bi) z  ⇒  z(a – 1) + biz  =  az + bzi 
=  z   (a – 1) + bi  =  z   a + bi 
⇒(a – 1)^{2} + b^{2} = a^{2} + b^{2} ∴ a = 1/2
since  a + bi  = 8
= a^{2} + b^{2} = 64
∴ u = 255 & v = 4 ⇒ u + v = 259
Ex.18 Show that
Sol.
We have (5 + i) = 26 (cosθ + isinθ), where tanq = 1/5 and therefore (5 + i)^{4} = 676(cos^{4}θ + isin ^{4}θ).
But (5 + i)4 = (24 + 10i) = 476 + 480i ; hence we have
cos 4θ = 476/676, sin 4θ = 480/676, and tan 4θ = 1, nearly.
∴ 4θ = π/4 approximately.
Ex.19 If a & b are complex numbers then find the complex numbers z_{1} & z_{2} so that the points z_{1} , z_{2} and a, b be the corners of the diagonals of a square .
Sol.
a  z_{1} = (b  z_{1}) e^{ i p/2} = i (b  z_{1})
a  i b = z_{1} (1  i)
Ex.20 Find the square root of
Sol.
Ex.21 On the Argand plane point 'A' denotes a complex number z_{1} . A triangle OBQ is made directly similar to the triangle OAM, where OM = 1 as shown in the figure . If the point B denotes the complex number z_{2}, then find the complex number corresponding to the point 'Q' in terms of z_{1} & z_{2} .
Sol .
= ∠ BOM  ∠ AOM = ∠ BOM  ∠ BOQ = ∠ QOM = amp of z ( ∠ AOM = ∠ BOQ = θ) Hence complex number corresponding to the point Q = Z_{2}/Z_{1}
Ex.22 Compute the product
Sol. Assume multiply numerator and denominator by (1  z) which simplifies to
Ex.23 Find the set of points on the complex plane such that z^{2} + z + 1 is real and positive (where z = x + i y) .
Sol. x^{2}  y^{2} + 2xy i + x + i y + 1 is real and positive ⇒ (x^{2}  y^{2} + x + 1) + y (2x + 1) i is real and positive ⇒ y (2x + 1) = 0 and x^{2}  y^{2} + x + 1 > 0 if y = 0 then x^{2} + x + 1 is always positive
complete x  axis if x = 1/2 then 3/4  y^{2} >0
156 videos176 docs132 tests

156 videos176 docs132 tests
