argand diagrams

Question 1

modulus |z| where z = x + iy

Answer 1

|z| = √(x² + y²)
is the distance from the origin to z on an argand diagram

Question 2

arg z where z = x + iy

Answer 2

tanθ = y/x
θ is the argument, shows the angle between the positive real axis and the line joining z to the origin

Question 3

if μ is the positive acute angle made with the real axis by the line joining the origin and z

Answer 3

first quadrant: arg z = μ
second quadrant: arg z = π - μ
third quadrant: arg z = -(π - μ)
fourth quadrant: arg z = -μ

Question 4

modulus argument form of z

Answer 4

z = r(cosθ + isinθ)

|z| = r
arg z = θ

Question 5

formulas connecting any z1 and z2

Answer 5

|z1 × z2| = |z1||z2|
arg(z1 × z2) = arg z1 + arg z2
|z1 ÷ z2| = |z1| ÷ |z2|
arg(z1 ÷ z2) = arg z1 - arg z2

Question 6

how to find difference between z1 and z2

Answer 6

|z2 - z1|

Question 7

if z = x + iy is a locus of points

Answer 7

z1 = x1 + iy1 is the locus of points (centre of circle)
|z - z1| = r
is a circle with centre (x1, y1) and radius r

Question 8

how find perpendicular bisector of two lines z1 and z2

Answer 8

|z - z1| = |z - z2|
joins z1 and z2

Question 9

how to use z1 with argument

Answer 9

arg(z - z1) = θ

z1 is the fixed point
θ is the angle with a line from the fixed point which is parallel to the real axis