Further Pure 1

1
Further Pure 1

• Complex Numbers
• LOCI

2
Sets of points in Argand diagram

• What does z2 z1 represent?
• If z1 x1 y1j
• z2 x2 y2j
• Then z2 z1
• (x2 x1) (y2 y1)j
• So z2 z1
• v((x2 x1)2 (y2 y1)2)
• This represents the distance between to complex
  numbers
• z1 z2.

Im
(x2,y2)
y2- y1
(x1,y1)
x2- x1
Re
3
Example 1

• Draw an argand diagram showing the set of points
  for which z 3 4j 5
• Solution
• First re-arrange the question
• z (3 4j) 5
• From the previous slide this represents a
  constant distance of 5 between the point (3,4)
  and z.
• This will give a circle centre (3,4)
• Now do Ex 2c q1

4
Sets of points using the Modulus-Argument form

• What do you think arg(z1-z2) represents?
• If z1 x1 y1j z2 x2 y2j
• Then z2 z1 (x2 x1) (y2 y1)j
• Now arg(z2 z1) inv tan ((y2 y1)/(x2 x1))
• This represents the angle between the line from
  z1 to z2 and a line parallel to the real axis.
• So arg(z2 z1) a

Im
(x2,y2)
y2- y1
a
(x1,y1)
x2- x1
Re
5
Example 2

• Draw Argand diagrams showing the sets of points z
  for which
• i) arg z p/3
• ii) arg (z - 2) p/3
• iii) 0 arg (z 2) p/3
• Now do Ex 2c q2

6
Example 3

• Sketch the locus of points z 2 z i.
• Describe in words what this means.
• z 2 is the distance from the co-ordinate
  A(-2,0) to any point z x iy. z 2 z
  (-2)
• z i is the distance from the co-ordinate
  B(0,1) to any point z x iy.
• As the position of z is the same then this means
  the distance from z to (-2,0) must be the same as
  the distance from z to (0,1).

7
Example 3

• In other words z is equidistant from bath
  co-ordinates.
• This is just like a perpendicular bisector from
  GCSE.
• Ex 2c q3

Im
(0,1)
(-2,0)
Re
8
Example 4

• Sketch the locus of z such that z 3
  2z 1 i
• So z 3 2z (1 i)
• What does this locus mean?
• This will be the locus of points (x,y) such that
  the distance between (3,0) and (x,y) is twice the
  distance between (1,-1) and (x,y).
• Here its not as obvious to spot what the locus
  will look like so you need to use algebra.
• You could have done this for the previous example
  if you did not spot the line.

9
Example 4
10
Example 4

• The locus of points is a circle centre (1/3,-4/3)
  with a a radius sqrt(20/9).
• Now prove the previous example using algebra.
• You should get the equation as 4x 2y 3 0
• Ex 2c q 4

11
Example 5

• Sketch the Locus of z such that
• Remember that arg(a/b) arg(a) arg(b)
• So
• First draw arg(z-2) and arg(z5) on a diagram

C(x,y)
A(-5,0 )
B(2,0)
12
Example 5

• Now let arg(z5) a and arg(z-2) ß.
• Lets call the angle at C, ?.
• Remember we want
• arg(z-2) arg(z5) p/4
• So we want
• ß a p/4
• Now
• a ? ß, so ? p/4.
• Now we have the locus of points where ? must p/4.

C(x,y)
?
ß
a
A(-5,0 )
B(2,0)
13
Example 5

• From circle Theorems at GCSE the angle inside a
  segment taken from either end of a chord is
  always the same.
• The locus of points must be an arc of a circle.

C(x,y)
?
ß
a
A(-5,0 )
B(2,0)
14
Example 5

• From circle Theorems at GCSE the angle inside a
  segment taken from either end of a chord is
  always the same.
• The locus of points must be an arc of a circle.
• Ex 2c q5

?
?
?
?