Edexcel A Level Further Maths: Core Pure:复习笔记1.1.5 Loci in Argand Diagrams
Loci in Argand Diagrams
How do I sketch the locus of
or
on an Argand diagram?
- All complex numbers,
, that satisfy the equation lie on a vertical line with Cartesian equation 
- Any complex number along this vertical line will have a real part of
- Any complex number along this vertical line will have a real part of
- All complex numbers, , that satisfy the equation lie on a horizontal line with Cartesian equation
- Any complex number along this horizontal line will have an imaginary part of
- Any complex number along this horizontal line will have an imaginary part of
- E.g. The loci
and 
are represented by the vertical line 
and the horizontal line 
Sketching the loci of and
How do I sketch the locus of
on an Argand diagram?
- All complex numbers,
, that satisfy the equation 
lie on a circle of radius about the origin
- E.g. the locus of
is a circle of radius 10, centred at the origin, as every complex number on that circle has a modulus of 10
- E.g. the locus of
- For a given complex number,
, all complex numbers, , that satisfy the equation lie on a circle of radius about the centre
- This is because
represents the distance between complex numbers and - E.g. the locus of
is a circle of radius 10 about 
- This is because
- Many equations need to be adjusted algebraically into the correct form
- E.g. to find the centre of the circle
, first rewrite it as
, giving the centre as
- E.g. to find the centre of the circle
, first rewrite it as 
, giving the centre as （-i） - Note that the centre of the circle
is the origin (it can be thought of as 
- E.g. to find the centre of the circle
- In order to sketch correctly, check whether the origin lies outside, on or inside the circle
- E.g. for the locus of
the distance from the centre of the circle, 
, to the origin is 5 (by Pythagoras), which is less than the radius of 10; a sketch must therefore show the origin inside the circle
- E.g. for the locus of
- By knowing the radius and centre of a circle, the Cartesian equation of the circle can be found
- The circle
has a radius of 10 and centre of （3，4） in coordinates, so the equation of the circle is 
- The circle
Sketching the loci ofand
How do I sketch the locus of 
on an Argand diagram?
- For two given complex numbers,
and 
, all complex numbers, 
, that satisfy the equation lie on the perpendicular bisector of and
- This is because the distance from to must equal the distance from to
- a condition that is satisfied by all the complex numbers, , on the perpendicular bisector of and
- a condition that is satisfied by all the complex numbers,
- E.g. the locus of
can be rewritten as 
which is the perpendicular bisector of the points 
and -8
- This is because the distance from
- A sketch of the perpendicular bisector is sufficient, without finding its exact equation (though this could be found using coordinate geometry methods)
Sketching the loci of 
and
How do I sketch the locus of 
on an Argand diagram?
- All complex numbers, , that satisfy the equation
lie on a half-line from the origin at an angle of 
to the positive real axis
- Although the half-line starts at the origin, the origin itself (
) does not satisfy the equation as
is undefined (there is no angle at the origin) - To show the exclusion of from the locus of, a small open circle at the origin is used
- E.g. the locus of
is a half-line of angle 
to the positive real axis, starting from the origin, with an open circle at the origin
- Although the half-line starts at the origin, the origin itself (
- For a given complex number, , all complex numbers, , that satisfy the equation lie on a half-line from the point at an angle of to the positive real axis, with an open circle to show the exclusion of
- E.g. the locus of
can be rewritten as 
, which is a half-line of angle 
measured from the point 
, with an open circle at to show its exclusion
- E.g. the locus of
- In some cases, the equation of the half-line can be found using a sketch to help
- E.g. the locus of is the half-line
- E.g. the locus of
can be thought of, in coordinate geometry, as the half-line through （8，5）with gradient -1, giving 
- Whilst not examinable, the half-line equation for a more general angle,, is
, as the 
- E.g. the locus of
Sketching the loci ofand
Exam Tip
- In the exam, do not worry about making your diagrams perfect.
- A quick sketch with all the key features is sufficient.
Worked Example
On separate axes, sketch the locus of points representing complex numbers,, that satisfy the following equations:
a)
b)
c)
d)
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