Important Results in Context with Rotation
(i) arg z = 0 represents points (non-zero) on a ray emanating from origin making an angle θ with positive direction of real axis.
(ii) arg (z-z1)= 0 represents points ( ≠3,) on ray emanating from z, making an angle θ with positive direction of real axis.
Rotation theorem:
(i) If P(z1) and Q(z2) are two complex numbers such that |z1| = |z2|, then z2=z1 eiθ, where θ = ⁄POQ.
(ii) If P(z1), Q(z2) and R(z3) are three complex numbers and ⁄PQR=Q, then
(iii) If P(z1), Q(z2), R(z3) and S(z4) are three complex numbers and ⁄STQ= θ, then
2. amp (z) = θ is a ray emanating from the origin inclined at an angle θ to the x-axis.
3. |z-a|=|z-b| is the perpendicular bisectors of the line joining a to b.
4. The equation of a line joining z1 and z2 is given by z = z1 + t(z1-z2) where t is a real parameter.
5. z = z1 (1+it) where t is a real parameter is a line through the point z, and perpendicular to the line joining z, to the origin.
6. The equation of a line passing through z1 and z2 can be expressed in the determinant form as
This is also the condition for three complex numbers z, z1, z2 to b collinear.
7. The equation of a circle having centre z0 and radius p is:
|z-z0| = p or z
- z0- 0 - z + 0z0 - p2 = 0 which is of the form
z + 
z + α + k = 0, k is real.
Centre is -α and radius =√(α - k) circle will be real if α - k ≥ 0.
8. The equation of the circle described on the line segment joining z1 and z2 as
diameter is arg (z-z2)/(z-z2 )= ±Π/2 or (z-z1) (-2)+(z-z2 )(1)=0
9. Condition for four given points z1, z2, z3 and z4 to be concyclic is the number
(z3 z1)/(z3-z2 ).(z4 z2)/(z4-z1 ) should be real.
Hence, the equation of a circle through 3 non-collinear points z1, z2 and z3 can be taken as
(z-z1 )(z3-z1 )/((z-z1 )(z3-z2)) is real
=> (z-z2 )(z3-z1 )/((z-z1 )(z3-z2))=(-2 )(3- 1)/((z-z1 )(z3-z2))
10. Arg((z-z1)/(z-z2)) = 0 represent
(i) a line segment if θ =Π
(ii) pair of ray if θ = 0
(iii) a part of circle, if 0<0<Π
11. Area of triangle formed by the points z1, z2 and z3 is
12. Perpendicular distance of a point z0 from the line
z + α + r = 0 is |( z0 + α + r )/α|α||
13. (i) Complex slope of a line z + α + r = 0 w = - α/α
(ii) Complex slope of a line joining the points z1 and z2 is w=(z1-2)/(z1-2)
(iii) Complex slope of a line making Q angle with real axis w=l2iθ
14. Dot and Cross product
Let z1 = x1 + i y1 and z2 = x2 + i y2 be two complex numbers (vectors). The dot product (also called the scalar product) of z1 and z2 is defined by z1.z2 = |z1| |z2| cos θ
= x1x2 + y1y= = Re {1z2}
= 1/2{1z2 + z12}
Where θ is the angle between z1 and z2 which lies between 0 and Π .
If vectors z1, z2 are perpendicular then z1.z2 = 0 => z1/z2 + z2/z2 = 0
i.e. sum of complex slopes = 0
The cross product of z1 and z2 is defined by
z1 × z2 = |z1| |z2| sin θ|
= x1y2 - y1x2
= Im{1z2 }
= 1/zi (z2 - z12)
If vectors z1, z2 are parallel then z1 × z2 = 0 z1/z2 = z2/z2
i.e., complex slopes are equal.
Note:
w1 and w2 are complex slopes of two lines.
(i) If lines are parallel then w1 = w2
(ii) If lines are perpendicular then w1 + w2 = 0
15. If |z-z1| + |z-z2| = K> |z1-z2| then locus of z is an ellipse focii are z1 and z2.
16. If |z-z0| = |z + α + r/2K1| then locus of z is parabola whose focus is z0 and directrix is the line
z + α + r = 0 (provided z0+ α0 + r ≠ 0)
17. If |(z-z1)/(z-z2)| = k < |z1-z2| then locus of z is a circle
18. If then locus of z is a hyperbola, whose focii are z1 and z2.
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