1. Complex numbers
Loci

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

3. Considering a complex number as a vector
WB10 Loci using the argument
Considering a complex number as a vector
If you are on this line,
the angle and therefore argument remains constant
Hence the statement
describes the half-line from z1 at an angle α with the positive real axis

4. WB11 sketch the locus of z when arg 𝑧−(2−𝑖) = 𝜋 3
describes the half-line from point 𝑧 1 2, −1 at an angle 𝜋 3 with the positive real axis
locus of z

5. WB12 the locus of an arc locus of z If and then but and if then
How can z vary but keep θ fixed?
Circle theorem: angles in the same segment are equal
Hence the locus of z when is an arc passing through z1 and z2
such that the angle subtended by the chord between z1 and z2 on the arc is θ

6. → arg 𝑧−(0+2𝑖) 𝑧+(3+0𝑖) = 𝜋 4 𝑧 1 =2𝑖, 𝑧 2 =−3 → arg 𝑧−2𝑖 𝑧+3 = 3𝜋 4
WB a) sketch the locus of z when arg 𝑧−2𝑖 𝑧+3 = 𝜋 4
Put in the form arg 𝑧− 𝑧 1 𝑧− 𝑧 to identify z1 and z2
→ arg 𝑧−(0+2𝑖) 𝑧+(3+0𝑖) = 𝜋 4
locus of z
𝑧 1 =2𝑖,
𝑧 2 =−3 →
b) What would be the equation of the locus of the minor arc?
→ arg 𝑧−2𝑖 𝑧+3 = 3𝜋 4
Circle theorem: aopposite angles in a cyclic quadrilateral add to π (180)

7. WB14 generalising / how do we know which side to put the arc?
e.g. sketch the locus of z when arg 𝑧−2𝑖 𝑧+3 =θ, θ>0
As in WB13
𝑧 1 =2𝑖,
𝑧 2 =−3
locus of z
Therefore locus is correct
So locus is incorrect
locus of z ?

8. 𝑧 1 =5, 𝑧 2 =1 Put in the form arg 𝑧− 𝑧 1 𝑧− 𝑧 2 to identify z1 and z2
WB15 a) sketch the locus of z when arg 𝑧−5 𝑧−1 = 𝜋 2
Put in the form arg 𝑧− 𝑧 1 𝑧− 𝑧 to identify z1 and z2
𝑧 1 =5,
𝑧 2 =1 →
locus of z
b) Find the centre of the resulting locus
If 𝜃= 𝜋 2 then the chord between z1 and z2
is a diameter
The centre of the circle is (3,0)

9. WB16 given that arg 𝑧−2𝑖 𝑧+2 = 𝜋 2 a) sketch the locus of P
b) deduce the value of 𝑧+1−𝑖
𝑧 1 =2𝑖,
𝑧 2 =−2 →
Put in the form arg 𝑧− 𝑧 1 𝑧− 𝑧 to identify z1 and z2
also θ= 𝜋 2 so the locus is a semicircle
And the chord between z1 𝑎𝑛𝑑 z2 is a diameter
so locus is correct
b) 𝑧+1−𝑖 is the distance from a
point on the locus to the point (-1,1)
But (-1,1) is the centre of the circle,
So 𝑧+1−𝑖 must equal the radius of the circle
𝑧+1−𝑖 = 2

10. WB17 You may asked to identify a point satisfying two rules
find the complex number z which satisfies both 𝑧−3+2𝑖 =4
and 𝑎𝑟𝑔 𝑧−1 = 𝜋 4
First 𝑧−(3−2𝑖) =4 circle centre (3, -2) radius 4
The half-line from (1, 0) passes through the centre of the circle
As the angle is , the triangle is isosceles
Using Pythagoras,
𝑧− 𝑧 1 =𝑎 describes the circle
with centre z1 and radius a
𝑎𝑟𝑔 𝑧− 𝑧 1 =4 describes the half-line from 𝑧 1 at an angle  with the real axis

11. WB18 Max and Min values
The point P represents a complex number z in an Argand diagram.
Given that 𝑧+1−𝑖 =1
Find a Cartesian equation for the locus of P and sketch it in an Argand diagram b) Find the greatest and least values of 𝑧
c) Find the greatest and least values of 𝑧−1
𝑧+1−𝑖 = 𝑧−(1+𝑖) a circle centre(1, -1) radius 1
circle has Cartesian eqn (𝑥+1) 2 + (𝑦−1) 2 =1
is distance to origin
is distance to (1,0)

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

13. END