CIE A Level Maths: Pure 3复习笔记8.2.5 Inequalities & Regions in Argand Diagrams

How do I sketch inequalities such as on an Argand diagram?

• For a given complex number, the inequality represents all complex numbers,, that lie inside the circle of radius, centred at
• Here is how to represent the following inequalities as regions on an Argand diagram…
  • : shade the region inside the circle of radius , centred at
  • : shade the (unbounded) region outside circle of radius, centred at
  • Again use a dotted line for strict inequalities (< and >) and a solid line for weak inequalities ( ≤ or ≥ )

Sketching the inequalities and

How do I sketch inequalities such as on an Argand diagram?

• For two given complex numbers and, the inequality represents all complex numbers,, that lie closer to than to
• Here is how to represent the following inequalities as regions on an Argand diagram…
  • : shade the region on the side of the perpendicular bisector ofand that includes the point
  • : shade the region on the side of the dotted perpendicular bisector of and that includes the point
• A good way to remember which side to shade is that the inequality sign points (like an arrow) to the side to be shaded

Sketching the inequalities  

How do I sketch inequalities such as  on an Argand diagram?

• For a given complex number, , the inequality represents all complex numbers,, that have an argument between  and , as measured from the point
• Here is how to represent the following inequalities as regions on an Argand diagram…
  • : shade the wedge-shaped region between the half-line and the half-line , with an open circle at  to show the exclusion of this point (to avoid the undefined value of )

Sketching the inequalities

How do I draw multiple inequalities on the same Argand diagram?

• To sketch a region that satisfies multiple inequalities, we need to find the intersection of all the regions (where all regions overlap)
  • E.g. the region satisfied by  can be found as follows…
    • Using diagonal lines in the same direction, lightly shade the region (the outside of a circle of radius 2 about the origin)
    • Using diagonal lines in a different direction, lightly shade the region  (the inside of a circle of radius 5 about the origin)
    • Where the diagonal lines cross each other highlights the region satisfying both inequalities; this should be shaded boldly
  • E.g. the region satisfied by and can be found as follows…
    • Using diagonal lines in the same direction, lightly shade the region (the side including from the perpendicular bisector of the points and 1)
    • Using diagonal lines in a different direction, lightly shade the region (the wedge-shaped region from 0 to radians from the origin)
    • Where the diagonal lines cross each other highlights the region satisfying both inequalities; this should be shaded boldly

The inequalities on the left, the inequalitiesand on the right