Curves and regions (loci)

Identify and sketch loci in the Argand plane defined by modulus and argument conditions; interpret regions defined by modulus inequalities; express equations of circles and lines in complex form.

Worked examples

Identifying a circle from a modulus condition

Straightforward

Problem

Describe the locus

|z - 3 + 2i| = 4

and state its centre and radius.

Perpendicular bisector locus

Moderate

Problem

Describe and find the Cartesian equation of the locus

|z - 1 - 3i| = |z + 5 + i|

Region defined by a modulus inequality

Challenging

Problem

Sketch and describe the region

|z - 2 + i| < 3

, and find the range of values of

|z|

Practise

Q1·Straightforward

The locus

|z - (3 + 4i)| = 5

is a circle with centre

(a, b)

and radius

r

. State the value of

a

(the

x

-coordinate of the centre).

Q2·Straightforward

The locus

|z - (3 + 4i)| = 5

is a circle. State the radius.

Q3·Straightforward

Let

z = 5 + 12i

. Calculate

|z|

Q4·Straightforward

The locus defined by

\operatorname{Im}(z) = -2

is a horizontal line in the Argand plane. What is the

y

-coordinate of every point on this locus?

Q5·Moderate

The locus

|z - 2| = |z - 8|

is the perpendicular bisector of the segment joining

2 + 0i

and

8 + 0i

on the real axis. State the

x

-coordinate of every point on this locus.

Q6·Moderate

The locus

|z - 3i| = |z + 3i|

is the perpendicular bisector of the segment joining

3i

and

-3i

. State the

y

-coordinate of every point on this locus.

Q7·Moderate

A point

z

satisfies

\arg(z) = \dfrac{\pi}{4}

and

\operatorname{Re}(z) = 3

. Find

\operatorname{Im}(z)

Q8·Moderate

The circle

|z - 1 - 2i| = 3

has centre

(1, 2)

and radius

3

. What is the

y

-coordinate of the highest point on this circle?

Q9·Moderate

The circle

|z + 3 - 2i| = 5

has centre

(-3, 2)

and radius

5

. Find the positive

x

-coordinate where this circle crosses the real axis (where

\operatorname{Im}(z) = 0

). Give your answer to two decimal places.

Q10·Challenging

The locus

|z - 2 - 3i| \le 4

defines a circular region. Using the triangle inequality

|z| \le |z - w| + |w|

, find the maximum value of

|z|

for points in this region. Give your answer to two decimal places.

Q11·Challenging

The locus

|z - 4| = 2|z + 1|

is an Apollonius circle. By squaring both sides and expanding, show that this simplifies to the form

(x + a)^2 + y^2 = R^2

. State the radius

R

. Give your answer to two decimal places.

Q12·Challenging

The locus

|z - i| + |z + i| = 4

is an ellipse with foci at

i

and

-i

. Find the positive

x

-intercept of this ellipse (where

\operatorname{Im}(z) = 0

). Give your answer to two decimal places.

Open Math

Curves and regions (loci)

Complex numbers · MEX-12-02

Name:

Date:

Q1Straightforward

The locus

|z - (3 + 4i)| = 5

is a circle with centre

(a, b)

and radius

r

. State the value of

a

(the

x

-coordinate of the centre).

Q2Straightforward

The locus

|z - (3 + 4i)| = 5

is a circle. State the radius.

Q3Straightforward

Let

z = 5 + 12i

. Calculate

|z|

Q4Straightforward

The locus defined by

\operatorname{Im}(z) = -2

is a horizontal line in the Argand plane. What is the

y

-coordinate of every point on this locus?

Q5Moderate

The locus

|z - 2| = |z - 8|

is the perpendicular bisector of the segment joining

2 + 0i

and

8 + 0i

on the real axis. State the

x

-coordinate of every point on this locus.

Q6Moderate

The locus

|z - 3i| = |z + 3i|

is the perpendicular bisector of the segment joining

3i

and

-3i

. State the

y

-coordinate of every point on this locus.

Q7Moderate

A point

z

satisfies

\arg(z) = \dfrac{\pi}{4}

and

\operatorname{Re}(z) = 3

. Find

\operatorname{Im}(z)

Q8Moderate

The circle

|z - 1 - 2i| = 3

has centre

(1, 2)

and radius

3

. What is the

y

-coordinate of the highest point on this circle?

Q9Moderate

The circle

|z + 3 - 2i| = 5

has centre

(-3, 2)

and radius

5

. Find the positive

x

-coordinate where this circle crosses the real axis (where

\operatorname{Im}(z) = 0

). Give your answer to two decimal places.

Q10Challenging

The locus

|z - 2 - 3i| \le 4

defines a circular region. Using the triangle inequality

|z| \le |z - w| + |w|

, find the maximum value of

|z|

for points in this region. Give your answer to two decimal places.

Q11Challenging

The locus

|z - 4| = 2|z + 1|

is an Apollonius circle. By squaring both sides and expanding, show that this simplifies to the form

(x + a)^2 + y^2 = R^2

. State the radius

R

. Give your answer to two decimal places.

Q12Challenging

The locus

|z - i| + |z + i| = 4

is an ellipse with foci at

i

and

-i

. Find the positive

x

-intercept of this ellipse (where

\operatorname{Im}(z) = 0

). Give your answer to two decimal places.

Worked solutions and answers at openmath.au/year-12/extension-2/complex-numbers/curves-and-regions