# Worksheet: Loci in the Complex Plane Using the Modulus

In this worksheet, we will practice finding the loci of a complex equation in the complex plane from the modulus.

Q1:

A complex number satisfies .

Describe the locus of and give its Cartesian equation.

• AThe perpendicular bisector of the line segment between and , equation:
• BThe line joining and , equation:
• CThe perpendicular bisector of the line segment between and , equation:
• DThe circle with center and radius 1, equation:
• EIt is the line joining and , equation:

What is the minimum value of ?

• A
• B
• C
• D
• E

Q2:

The figure shows a locus of point in the complex plane, which is an ellipse with foci and and major axis . represents the number , the number , the number , and the number . Using the properties of an ellipse, write an equation for the locus of point in terms of . • A
• B
• C
• D
• E

Q3:

A complex number satisfies . Find the Cartesian equation of the locus of and describe it geometrically.

• A, a circle with center and radius 5
• B, the perpendicular bisector of the line segment joining and
• C, a circle with center and radius
• D, the perpendicular bisector of the line segment joining and
• E, a circle with center and radius

Q4:

Two loci are given by the equations and .

The complex numbers and satisfy both equations. Write these numbers in the form , where and are real numbers.

• A
• B
• C
• D
• E

Q5:

is a complex number such that is a real number. Find the Cartesian equation of the locus of point and describe it geometrically.

• A, a circle with center and radius
• B, a circle with center and radius
• C
• D, a straight line of slope and -intercept
• E, a straight line of slope and -intercept

Q6:

Consider point in the complex plane.

Find algebraically the locus of point such that .

• AIt is the line of the equation .
• BIt is the circle of the equation .
• CIt is the line of the equation .
• DIt is the line of the equation .
• EIt is the circle of the equation .

Which of the following is a correct geometrical description of the locus of point such that ?

• AIt is the circle with center and radius 1.
• BIt is the circle with center and radius 1.
• CIt is the perpendicular bisector of the line segment joining 1 and .
• DIt is the perpendicular bisector of the line segment joining 1 and .

Q7:

is a complex number such that is an imaginary number. Find the Cartesian equation of the locus of point and describe it geometrically.

• A, a circle with center and radius
• B, a circle with center and radius
• C, a straight line of slope and -intercept
• D, a straight line of slope and -intercept
• E, a circle with center and radius

Q8:

Describe the locus of such that and give its Cartesian equation.

• AA circle centered at with radius 5,
• BA circle centered at with radius 5,
• CA circle centered at with radius 5,
• DA circle centered at with radius 5,
• EA circle centered at with radius 5,

Q9:

Consider point in the complex plane.

Find algebraically the locus of point such that .

• AIt is the circle of the equation .
• BIt is the line of the equation .
• CIt is the line of the equation .
• DIt is the line of the equation .
• EIt is the line of the equation .

Which of the following is a correct geometrical description of the locus of point such that ?

• AIt is the perpendicular bisector of the line segment joining and .
• BIt is the circle with center and radius 3.
• CIt is the perpendicular bisector of the line segment joining and .
• DIt is the line joining and .
• EIt is the line joining and .

Q10:

Describe the locus of such that and give its Cartesian equation.

• AA circle centered at with radius 3,
• BA circle centered at with radius 3,
• CA circle centered at with radius 2,
• DA circle centered at with radius 3,
• EA circle centered at with radius 3,

Q11:

Describe the locus of such that and give its Cartesian equation.

• AA circle centered at the origin with radius 2,
• BA circle centered at the origin with radius 4,
• CA circle centered at the point with radius 2,
• DA circle centered at the origin with radius 4,
• EA circle centered at the origin with radius 2,

Q12:

Consider point in the complex plane.

Find algebraically the locus of point such that .

• AIt is the line of the equation .
• BIt is the line of the equation .
• CIt is the line of the equation .
• DIt is the circle of the equation .
• EIt is the line of the equation .

Which of the following is a correct geometrical description of the locus of point such that ?

• AIt is the perpendicular bisector of the line segment joining and .
• BIt is the perpendicular bisector of the line segment joining and .
• CIt is the circle with center and radius 4.
• DIt is the perpendicular bisector of the line segment joining and .
• EIt is the perpendicular bisector of the line segment joining and .

Q13:

A complex number satisfies .

Describe the locus of and give its Cartesian equation.

• AA circle centered at with radius 3,
• BA circle centered at with radius 3,
• CA circle centered at with radius 3,
• DA circle centered at with radius 3,
• EA circle centered at with radius 3,

Find the range of the argument of in the range .

• A
• B
• C
• D
• E

Find the range of the modulus of .

• A
• B
• C
• D
• E

Q14:

A complex number satisfies . Find the Cartesian equation of the locus of and describe it geometrically.

• A, a horizontal line with constant
• B, a circle with center and radius
• C, a circle with center and radius
• D, a circle with center and radius
• E, a circle with center and radius

Q15:

A complex number satisfies . Describe the locus of .

• AIt is an ellipse with foci at the origin and at and has a major axis of 4.
• BIt is an ellipse with foci at the origin and at and has a major axis of 4.
• CIt is an ellipse with foci at the origin and at and has a major axis of 8.
• DIt is an ellipse with foci at the origin and at and has a major axis of 8.
• EIt is a line with equation .

Q16:

A complex number lies at a distance of from and a distance of from . Does the point lie on the circle centered at the origin that passes through and ?

• ANo
• BYes

Q17:

The figure shows a locus of a point in the complex plane. Write an equation for the locus in the form , where is a constant to be found. • A
• B
• C
• D
• E

Q18:

The figure shows a circular locus of a point in the complex plane. Given that is the centre of the circle, write an equation for the locus in the form , where and are constants to be found. • A
• B
• C
• D
• E