# Worksheet: Loci in the Complex Plane Using the Argument

In this worksheet, we will practice finding the loci of a complex equation in the complex plane defined in terms of the argument.

Q1:

A half line is given by , . Write an equation for the half line in the form , where and are constants to be found.

• A
• B
• C
• D
• E

Q2:

Find the Cartesian equation of the locus of such that .

• A,
• B,
• C,
• D,
• E,

Q3:

Consider , , and in the complex plane.

Find the Cartesian equation of the locus of such that .

• A
• B
• C
• D
• E

Find the Cartesian equation of the locus of such that .

• A,
• B,
• C,
• D,
• E,

Where does the locus of meet the locus of ?

• AThe two loci do not meet.
• BAt and
• CAt
• DAt
• EAt and

Find the Cartesian equation of the locus of such that .

• A,
• B,
• C,
• D,
• E,

Where does the locus of meet the locus of ?

• AAt and
• BAt and
• CThe two loci do not meet.
• DAt
• EAt

Q4:

A half line is given by , . Write an equation for the half line in the form , where and are constants to be found.

• A
• B
• C
• D
• E

Q5:

Find the Cartesian equation of the locus of such that .

• A,
• B,
• C,
• D,
• E,

Q6:

Consider and in the complex plane.

Find the Cartesian equation of the locus of such that .

• A
• B
• C
• D
• E

Find the Cartesian equation of the locus of such that .

• A,
• B,
• C,
• D,
• E,

Find the point at which the two loci meet.

• A
• B
• C
• D
• EThe two loci do not meet.

Q7:

Find the Cartesian equation of the locus of such that .

• A,
• B,
• C,
• D,
• E,

Q8:

Which of the graphs shown is the correct representation of the locus of that satisfies ?

• A(d)
• B(e)
• C(b)
• D(a)
• E(c)

Q9:

Which of the graphs shown is the correct representation of the locus of that satisfies ?

• A(c)
• B(d)
• C(a)
• D(e)
• E(b)

Q10:

Which of the graphs shown is the correct representation of the locus of that satisfies ?

• A(a)
• B(c)
• C(d)
• D(b)
• E(e)

Q11:

Which of following figures is the correct representation of the locus of which satisfies ?

• Ae
• Bc
• Cd
• Da
• Eb

Q12:

Which of the following figures is the correct representation of the locus of which satisfies ?

• Ac
• Bd
• Ca
• De
• Eb

Q13:

The figure shows a locus of a point in the complex plane. Write an equation for the locus in the form , where and are constants to be found.

• A
• B
• C
• D
• E

Q14:

Consider and in the complex plane.

Find the Cartesian equation of the locus of such that .

• A,
• B,
• C,
• D,
• E,

Find the Cartesian equation of the locus of such that .

• A,
• B,
• C,
• D,
• E,

Find the point at which the two loci meet and the angle at which they meet.

• A,
• B,
• C,
• D,
• E,

Q15:

The point satisfies . By plotting the locus on an Argand diagram, find its Cartesian equation.

• A,
• B,
• C,
• D,
• E,

Q16:

Given that satisfies , by sketching the locus of , find the range of values of and the range of values of the principle argument of .

• A,
• B,
• C,
• D,
• E,

Q17:

The figure shows a locus of a point in the complex plane. Write an equation for the locus in the form , where and are constants to be found.

• A
• B
• C
• D
• E

Q18:

Find the Cartesian equation of the locus of which satisfies .

• A,
• B,
• C,
• D,
• E,

Q19:

Consider and in the complex plane.

Find the Cartesian equation of the locus of such that .

• A,
• B,
• C,
• D,
• E,

The locus of and the are tangent to a circle with radius 2 at the points and respectively. Find the coordinates of and .

• A,
• B,
• C,
• D,
• E,

Find the condition that must satisfy for the locus of point to trace out the arc of the circle with radius 2 between the points and counterclockwise.

• A
• B
• C
• D
• E

Q20:

Consider and in the complex plane.

Find the Cartesian equation of the locus of such that .

• A,
• B,
• C,
• D,
• E,

Find the Cartesian equation of the locus of such that .

• A,
• B,
• C,
• D,
• E,

Find the point at which the two loci meet and the angle at which they meet.

• A, angle:
• B, angle:
• C, angle:
• D, angle:
• E, angle:

Q21:

Find the Cartesian equation of the locus of such that .

• A,
• B,
• C,
• D,
• E,

Q22:

Sketch the locus of when .

• A
• B
• C
• D
• E