# 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 .

**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
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- 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