Loci
|z-a| = |z-b|
The locus represents the locus of all points which lie on the perpendicular bisector between the points represente
by the complex numbers a and b
be careful have to make sure the perpendicular bisector goes the right side of the origin.
finding the equation for the perpendicular bisector
treat the complex numbers like complex numbers
|z-3| = |z-i|
|x+iy-3| = |x+iy-i|
|(x-3)+iy| = |x+i(y-1)
root both sides and expand out
x^2 -6x +9 +y^2 = x^2 +y^2 - 2y +1
-6x+9 = -2y+1
y = 3x-4
Inequalities
How can we draw argand diagrams that show the following sets of points for z?
|z - 4 - i| < |z + 3 +2i|
we sketch the perpendicualr bisector as a dotted line cause we know its < not <=
can sub in a coordinate or read litterally what it is saying
the region is closer to < than it is to this one
Try the point 0 + 0i
LHS = | -4 -i | = sqrt 17
RHS = | 3 + 21| = sqrt 13
|z-a| = |z-b|
The locus represents the locus of all points which lie on the perpendicular bisector between the points represente
by the complex numbers a and b
be careful have to make sure the perpendicular bisector goes the right side of the origin.
finding the equation for the perpendicular bisector
treat the complex numbers like complex numbers
|z-3| = |z-i|
|x+iy-3| = |x+iy-i|
|(x-3)+iy| = |x+i(y-1)
root both sides and expand out
x^2 -6x +9 +y^2 = x^2 +y^2 - 2y +1
-6x+9 = -2y+1
y = 3x-4
Inequalities
How can we draw argand diagrams that show the following sets of points for z?
|z - 4 - i| < |z + 3 +2i|
we sketch the perpendicualr bisector as a dotted line cause we know its < not <=
can sub in a coordinate or read litterally what it is saying
the region is closer to < than it is to this one
Try the point 0 + 0i
LHS = | -4 -i | = sqrt 17
RHS = | 3 + 21| = sqrt 13
