Edexcel A Level Further Maths: Core Pure:復習筆記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)
轉載自savemyexams
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