- #1

- 10

- 0

**Complex Locus - PLEASE HELP!!!!!!!!!!**

## Homework Statement

Q1: The complex number z satisfies arg(z+3) = pi/3

(a) Sketch the locus of the point P in the Argand diagram which represents z

**(DONE)**

(b) Find the modulus and argument of z when z takes its least value.

**(STUCK)**

(c) Hence represent z in a + ib form.

**(STUCK)**

## Homework Equations

arg(z+3) = pi/3

## The Attempt at a Solution

arctan[y/(x+3)] = pi/3

y/(x+3) = sqrt(3)

y = sqrt(3)x + 3*sqrt(3)

I can't do part (b) and hence (c) of Q1.

__________________________________________________________________

## Homework Statement

Q2: If z is any complex number such that |z| = 1, show using Argand diagram or otherwise that:

## Homework Equations

(a) 1 <= |z+2| <= 3

**(STUCK)**

(b) -pi/6 <= arg(z+2) <= pi/6

**(STUCK)**

## The Attempt at a Solution

I have sketched both parts (a) which is the region between the circles centred at (-2,0) and radii 1 and 3 and (b).

___________________________________________________________________

## Homework Statement

Q3: The complex number z = x+iy, x and y are real, such that |z - i| = Im(z).

(a) Show that the point representing z has a Cartesian Equation y = 1/2(x^2 + 1). Sketch the locus.

**(DONE)**

(b) Find the gradient of the tangent to this curve which passes through the origin. Hence find the set of possible values of the principal argument of z.

**(STUCK)**

## Homework Equations

|z - i| = Im(z)

y = 1/2(x^2 + 1)

## The Attempt at a Solution

for part (a)

sqrt[x

^{2}+ (y-1)

^{2}] = y

x

^{2}+ y

^{2}- 2y + 1 = y

^{2}

therefore, y = 1/2(x

^{2}+1)

Part (b) - NO IDEA!

_______________________________

Thanks for your help!!!