Loci and regions in the Argand diagram

The Argand diagram below shows a circle C

Write down the equation of the locus of C in the form

│ z – w │ = a

where w is a complex number whose real and imaginary parts are integers, and a is an integer.

It is given that z₁ is a complex number representing a point on C. Of all the complex numbers which represent points on C, z₁ has the least argument.

Find │z₁│

Give your answer in an exact form.

The diagram below shows a locus on an Argand diagram.

Which of the equations below represents the locus shown above?

Circle your answer.

The region R on an Argand diagram satisfies both z 2i 3 and arg (z)

Sketch R on the Argand diagram below.

Find the maximum value of z in the region R, giving your answer in exact form.

The complex number w is such that

It is given that w x iy, where x and y are real and x 0

Find an equation for y in terms of x

The complex number z satisfies both

The region R is the locus of z

Sketch the region R on the Argand diagram below.

z₁ is the point in R at which z is minimum.

Calculate the exact value of z₁

Express z₁ in the form a ib, where a and b are real.

The complex number z satisfies the equations

and

where a is real.

numbers.

A circle C in the complex plane has equation z 2 5i a

The point z₁ on C has the least argument of any point on C, and arg (z₁)