# 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