1. Complex numbers
Loci

2. Complex numbers: Loci in the Argand diagram
KUS objectives
BAT Use complex numbers to represent a region of points on an Argand diagram
Starter:

3. Reminder: 6 types of transformations of graphs
Sketch or describe the six types of transformation of 𝑦=𝑓(𝑥) represented by:
𝒇(𝒙+𝑨) 𝒇 𝒙 +𝑨 −𝒇(𝒙)
𝑨𝒇(𝒙) 𝒇(𝑨𝒙) 𝒇(−𝒙)

4. WB21 what is the effect of the transformation 𝑚= 𝑥 2 on the function 𝑦= 𝑥 2 ?
In GCSE you learnt that 4 𝑓(𝑥) was a vertical stretch x 4
Taking a slightly different approach to this transformation
Rearranging:
Substituting this:
Graphically
This method can be used to understand transformations in the complex plane

5. w=𝑘𝑧 make z the subject 𝑧= 𝑤 𝑘 So 𝑧 = 𝑤 𝑘 =𝑎 𝑤 𝑘 =𝑎 So 𝑤 =𝑎 𝑘 =𝑎𝑘
WB22 what is the effect of the transformation w=𝑘𝑧 on the locus
given by 𝑧 =𝑎 Where k is real and positive
w=𝑘𝑧 make z the subject
𝑧= 𝑤 𝑘
substitute into the locus expression
locus of z
So 𝑧 = 𝑤 𝑘 =𝑎
locus of w
Rearrange to
𝑤 𝑘 =𝑎
So 𝑤 =𝑎 𝑘 =𝑎𝑘
So the transformation is a stretch factor k

6. Substitute into locus expression & rearrange to a familiar locus
WB23 what is the effect of the transformation w=𝑧+ 𝑧 1 on the locus
given by 𝑧 =𝑎 Where k is real and positive
Make z the subject
locus of w
Substitute into locus expression & rearrange to a familiar locus
locus of z
So 𝑤− 𝑧 1 =𝑎 is a circle centre 𝑧 1 radius a

7. Substitute into locus expression & rearrange to a familiar locus
WB24 what is the effect of the transformation w=𝑘𝑧+ 𝑧 1 on the locus
given by 𝑧 =𝑎 Where k is real and positive
Make z the subject
locus of w
Substitute into locus expression & rearrange to a familiar locus
locus of z
So 𝑤− 𝑧 1 =𝑘𝑎 is a circle centre 𝑧 1 radius ka

8. Typically this requires introducing 𝑤=𝑢+𝑖𝑣 and some manipulation
Summary so far
w = z + z1 where z1 = a + ib represents a translation vector 𝑎 𝑏
w = kz represents an enlargement scale factor k centre (0,0)
w = kz + z1 where z1 = a + ib represents an enlargement scale factor k centre (0,0),
followed by a translation vector 𝑎 𝑏
Other transformations are less obvious to spot, but can be interpreted using the same method as before
Make z the subject
Substitute into locus expression & rearrange to a familiar locus
Typically this requires introducing 𝑤=𝑢+𝑖𝑣 and some manipulation

9. WB25 a) Show that the transformation 𝑤= 𝑧−1 𝑧 maps 𝑧−1 =1 in the
z-plane onto 𝑤 = 𝑤−1 in the w-plane
The region 𝑧−1 ≤1 in the z-plane is mapped onto the region T in the w-plane
b) Shade the region T on an argand diagram
a) Make z the subject
Sub in
is perpendicular bisector between (0,0) and (1,0)
Making common denominator
Simplifying so
Cross-multiplying
is left of this line

10. WB26 The transformation T from the z-plane to the w-plane is given by 𝑤= 3𝑧−2 𝑧+1
Show that the image, under T, of the circle with equation 𝑥 2 + 𝑦 2 =4 in the z-plane is a circle C in the w-plane. State the centre and radius of C.
Circle 𝑥 2 + 𝑦 2 =4 is represented by 𝑧 =2
Use 𝑤=𝑢+𝑖𝑣
Circle centre , radius

11. WB27 A transformation T of the z-plane to the w-plane is given by 𝑤= 𝑖𝑧−2 1−𝑧
Show that as z lies on the real axis in the z-plane, then w lies on the line L in the w-plane. Sketch L on an Argand diagram
So 𝑧=𝑥+0𝑖 as z lies on the real axis
Use 𝑤=𝑢+𝑖𝑣
As z can be anywhere on the real axis and so its locus is z = x + 0i, we need to manipulate our expression into this form, starting by using the complex conjugate in order to separate real and imaginary parts. We can then equate them with the known expression for z = x + 0i
locus of w

12. WB28a The transformation T from the z-plane to the w-plane is given by 𝑤= 𝑧+𝑖 𝑧
The transformation T maps the points on the line with equation 𝑦=𝑥 in the z-plane to points on the line L in the w-plane. Find an equation of L
Show that the image, under T, of the line with equation 𝑥+𝑦+1=0 in the z-plane is a circle C in the w-plane. State the centre and radius of C.
Sketch l and C on the same Argand diagram
locus of w
Use

13. WB28bc The transformation T from the z-plane to the w-plane is given by 𝑤= 𝑧+𝑖 𝑧
The transformation T maps the points on the line with equation 𝑦=𝑥 in the z-plane to points on the line L in the w-plane. Find an equation of L
Show that the image, under T, of the line with equation 𝑥+𝑦+1=0 in the z-plane is a circle C in the w-plane. State the centre and radius of C.
Sketch l and C on the same Argand diagram
b) If z lies on 𝑥+𝑦+1=0 C L
Circle centre , − 1 2
radius

14. WB29 The transformation T from the z-plane to the w-plane is defined
by 𝑤= 2(1+𝑖) 𝑧+2 Show that the locus of P in the z-plane is mapped to part of a straight line in the w-plane, and show this in an Argand diagram
P is the locus of z such that , found previously
Make z the subject
Sub in
‘Part of a line’ suggests keep in arg( ) form
is a half-line starting from w at an angle of 90o to (1,0)

15. WB30
June 11 Q5 – pure evil
The point P represents the complex number z on an Argand diagram, where the locus of P as z varies is the circle C with Cartesian equation 𝑥 2 + (𝑦−1) 2 =4
The locus of P as z varies is the curve C.
A transformation T from the z-plane to the w-plane is given by 𝑤= 𝑧+𝑖 3+𝑖𝑧 , 𝑧≠3𝑖
The point Q is mapped by T onto the point R. Given that R lies on the real axis,
(c) show that Q lies on C.
Most questions ask you to transform z onto w, this question does the opposite
Hence you want to consider an expression for w in terms of z = x + iy
R on x-axis

16. The perfect formula? Im Re When and in the exponential form this gives
But if
This gives Re
This single formula brings together key concepts in mathematics – integers, negative numbers, pi, exponentials, real and complex numbers!
Complex numbers have many applications in Mathematics, science and engineering.
What started as an ‘imaginary’ concept actually turns out to have many real applications such as fluid dynamics, electrical engineering, control systems in mechanical systems, quantum mechanics, relativity and differential equations in areas as far reaching as economics and biology.

17. KUS objectives
BAT Use complex numbers to represent a region of points on an Argand diagram
self-assess
One thing learned is –
One thing to improve is –

18. END