1. Complex Number:
A number of the z = x+iy (x, y Î R, I =Ö-1 is called a complex number, where x is called a real part i.e. x= Re(z) and y is called an imaginary part i.e. y = Im(z)
Modulus |z| = Ö(x2 + y2),
amplitude or amp (z) = arg(z) = q = tan-1 (y/x).
(i) Polar representation:
x = r cosq, y=sinq, r= |z| = Ö(x2 +y2)
(ii) Exponential form:
Z= reiq, where r =|z|, q = amp.(z)
(iii) Vector representation:
P(x,y) then its vector representation is z=OP
2. Integral Power of Iota:
i = Ö-1, i2 = -1 i3 = - i, i4 = 1
Hence i4n+1 = i, i4n+2 = -1, i4n+3 = - i, i4n or i4(n+1) = 1
3. Complex conjugate of z:
If z= x + iy, then `z = x –iy is called complex conjugate of z
* `z is the mirror image of z in the real axis.
* |z| = |`z|
* z + `z = 2Re(z) = purely real
* z - `z = 2i Im(z)= purely imaginary
* z`z = |z|2
* 
* 
* ` (z1z2) = `z1 `z2
* 
* `(zn) = (`z)n
* `(`z) = z
* If a = f(z), then `a = f(`z)
where a = f(z) is a function in a complex variable with real coefficients.
* z + ` = 0 or z = -`z Þ z = 0 or z is purely imaginary
* z = `z Þ z is purely real
4. Modulus of a complex number:
Magnitude of a complex number z is denoted as |z| and is defined as
(i) z`z = |z|2 = |`z|2
(ii) z-1 = (`z/|z|2 )
(iii) |z1 ± z2|2 + |z1|2 + |z2|2 ± Re (z1 `z2)
(iv) |z1 +z2|2 + |z1 -z2|2 = 2[|z1|2 + |z2|2]
(v) |z1 ± z2| £ |z1| + |z2|
(vi) |z1 ± z2| ³ |z1| - |z2|
5. Argument of a complex number:
Argument of a complex number z is the Ð made by its radius vector with + ve direction of axis.
arg z = q, z Î 1st quad.
= p - q, z Î 2nd quad.
= - q, z Î 3rd quad.
= q - p, z Î 4th quad.
(i) arg (any real + ve no.) = 0
(ii) arg (any real – ve no.) = p
(iii) arg (z-`z) = ± p/2
(iv) arg (z1.z2) = arg z1 + arg z2 + 2kp
(v) arg (z1 / z2) = arg z1 - arg z2 + 2kp
(vi) arg (`) =-arg z = arg (1/z), if z is non real = arg z, if real
(vii) arg (-z) = arg z + p, arg z Î (-p,0) = arg z - p, arg z Î (0, p)
(viii) arg (zn) = n arg z+2kp
(ix) arg z + arg `z = 0 argument function behaves like log function.
6. Square root of a complex no.
7. De-Moiver’s Theorem:
It states that if n is rational number, then
(cosq + isinq)2 = cosq + isin nq
8. Euler’s formulae as z = reje = cosq - I sinq
\ eje + e-je = 2cosq and ejq - e-jq = 2 I sinq
9. nth roots of complex number z1/n:
where m = 0, 1, 2, ……. (n-1)
(i) sum of all roots of z1/n is always equal to zero
(ii) product of all roots of z1/n = (-1)n-1 z
10. Cube root of unity:
Cube root of unity are 1, w, w2 where w = ((-1+iÖ3)/2) and 1+w+w2 = 0, w3 = 1
11. Some important result:
If z =cosq + isinq
(i) z+(1/z) = 2cosq
(ii) z-(1/z) = 2isinq
(iii) zn+(1/zn) = 2cosnq
(iv) If x=cosa +isina , y= cosb + I sinb & z = cosg + ising and given x + y + z =0, then
(a) (1/x) + (1/y) + (1/z) = 0
(b) yz +zx + xy =0
(c) x2 + y2 + z2 = 0
(d) x3 + y3 + z3 = 3xyz
12. Equation of Circle:
* |z-z1| = r represents a circle with centre z1 and radius r.
* |z| = r represents a circle with centre at origin.
* |z-z1| < r and |z-z1| > r represents interior and exterior of circle |z-z1| =r
* z`z + a`z +`az + b =0 represents a general circle where a Î c and b Î R.
* Let |z| = r be the given cirde,then equation of tangent at the point z1 is z`z1 + z`z1 = 2r2
* diametric form of circle:
or |z-z1|2 + |z-z2|2 = |z1-z2|2 where z1, z2 are end point of diameter and z is any point on circle.
13. Some important points:
(i) Distance formola PQ = |z2-z1|
(ii) Section formula
For internal division =
For external division =
(iii) Equation of straight line.
* parametric form z = tz1 + (1-t)z2 where t Î R
* Non parametric form
* Three points z1,z2,z3 are collinear if
or slope of AB = slope of BC = slope of AC.
(iv) The complex equation |(z-z1)/(z-z2| = k represents a circle if k ¹ 1 and a straight line if k =1
(v) The triangle whose vertices are the points represented by complex number z1, z2, z3 is equilateral if
i.e. if z21 + z22 + z23 = z1z2 + z2z3 + z1z3.
(vi) |z-z1| = |z-z2|=l represents an ellipse if |z1-z2| < l, having the point z1 and z2 as its foci and if |z1-z2| < l, then z lies on a line segment connecting z1 & z2
(vii) |z-z1| ~ |z-z2|= l represents a hyperbola if |z1-z2| > l, having the point z1 and z2 as its foci and if |z1-z2| = l, then z line on the line passing through z1 and a2 excluding the point between z1 & z2.
(viii) if four point z1, z2, z3, z4 are concyclic,
(ix) If three complex number are in A.P., then they lie on a straight line in the complex plane.
(x) If z1, z2, z3 be the vertices of an equilateral triangle and z0 be the circumcentre , than z21 + z22 + z23 = 3z20.
(xi) If z1, z2, z3 …….. zn be the vertices of a regular polygon of n sides & z0 be its centroid, then z21 + z22 + …….. + z2n = nz20.
(xii) If z1, z2, z3 be the vertices of a triangle, then the triangle is equilateral
Iff (z1-z2)2 + (z2-z3)2 + (z3-z1)2 = 0
(xiii) If z1, z2, z3 are the vertices of an isosceles triangle right angled at z2,
Then z21 + z22 + z23 = 2z2 (z1+z3).
(xiv) z1, z2, z3, z4 are vertices of a parallelogram then z1 + z3 = z2 + z4
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