Loci and regions in the Argand diagram

z₁ a bi and z₂ c di

where a, b, c and d are real constants.

Given that

• z₁ z₂ is real

show that a 2 and determine the value of each of b, c and d

On the same Argand diagram

• sketch the locus of points z which satisfy z 12 7

• sketch the locus of points w which satisfy w 5i 4

showing the coordinates of any points of intersection with the axes.

Given that

show that

Given that

determine the least value of z as z varies.

Given that

show that

Given that

determine the least value of z as z varies.

The locus of points z satisfies

| z + a i | = 3 | z – a | where a is an integer. The locus is a circle with its centre in the third quadrant and radius Determine

the value of a,

the coordinates of the centre of the circle.

The locus of points z = x + iy that satisfy

is an arc of a circle C.

On an Argand diagram sketch the locus of z.

Explain why the centre of C has x coordinate 5

Determine the radius of C.

Determine the y coordinate of the centre of C.

The point P is one vertex of a regular pentagon in an Argand diagram. The centre of the pentagon is at the origin.

Given that P represents the complex number 6 + 6i, determine the complex numbers that represent the other vertices of the pentagon, giving your answers in the

On a single Argand diagram, shade the region, R, that satisfies both

Determine the exact area of R, giving your answer in simplest form.

The point P is one vertex of a regular pentagon in an Argand diagram. The centre of the pentagon is at the origin.

Given that P represents the complex number 6 + 6i, determine the complex numbers that represent the other vertices of the pentagon, giving your answers in the

On a single Argand diagram, shade the region, R, that satisfies both

Determine the exact area of R, giving your answer in simplest form.

The point P in the complex plane represents a complex number z such that

Given that, as z varies, the locus of P is a circle,

determine the centre and radius of this circle.

Shade on an Argand diagram the region defined by the set

In an Argand diagram, the points A and B are represented by the complex numbers −3 + 2i and 5 − 4i respectively. The points A and B are the end points of a diameter of a circle C.

Find the equation of C, giving your answer in the form

The circle D, with equation |z− 2 − 3i| = 2, intersects C at the points representing the complex numbers z₁ and z₂

Find the complex numbers z₁ and z₂

In an Argand diagram, the points A and B are represented by the complex numbers −3 + 2i and 5 − 4i respectively. The points A and B are the end points of a diameter of a circle C.

Find the equation of C, giving your answer in the form

The circle D, with equation |z− 2 − 3i| = 2, intersects C at the points representing the complex numbers z₁ and z₂

Find the complex numbers z₁ and z₂

A transformation T from the z‑plane to the w‑plane is given by

The circle with equation z + i= 3 is mapped by T onto the circle C.

Show that the equation for C can be written as

Hence find

a Cartesian equation for C,

the centre and radius of C.

A transformation T from the z‑plane to the w‑plane is given by

The circle with equation z + i= 3 is mapped by T onto the circle C.

Show that the equation for C can be written as

Hence find

a Cartesian equation for C,

the centre and radius of C.

A complex number z = x + i y is represented by the point P in an Argand diagram.

Given that

show that the locus of P has equation

Hence find the maximum value of | z |