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 𝑧 satisfies | 𝑧 2 + 4 𝑖 | = | 𝑧 + 2 + 2 𝑖 | .

Describe the locus of 𝑧 and give its Cartesian equation.

  • AIt is the line joining 2 + 4 𝑖 and 2 + 2 𝑖 , equation: 𝑦 = 1 2 𝑥 + 3
  • BThe line joining 2 4 𝑖 and 2 2 𝑖 , equation: 𝑦 = 1 2 𝑥 3
  • CThe perpendicular bisector of the line segment between 2 + 4 𝑖 and 2 + 2 𝑖 , equation: 𝑦 = 2 𝑥 + 3
  • DThe perpendicular bisector of the line segment between 2 4 𝑖 and 2 2 𝑖 , equation: 𝑦 = 2 𝑥 3
  • EThe circle with center ( 2 , + 4 ) and radius 1, equation: ( 𝑥 + 2 ) + ( 𝑦 4 ) = 1

What is the minimum value of | 𝑧 | ?

  • A 6 1 0 5
  • B 8 5 3
  • C 3 5 5
  • D 3 5 8
  • E 5

Q2:

The figure shows a locus of point 𝑧 in the complex plane, which is an ellipse with foci 𝑃 and 𝑄 and major axis 𝐴 𝐵 . 𝑃 represents the number 1 , 𝑄 the number 3 2 1 3 2 𝑖 , 𝐴 the number 1 2 + 2 𝑖 , and 𝐵 the number 4 2 1 4 2 𝑖 . Using the properties of an ellipse, write an equation for the locus of point 𝑧 in terms of 𝑧 .

  • A | | 𝑧 + 1 + 2 2 𝑖 | | + | | 𝑧 4 2 + 1 + 4 2 𝑖 | | = 1 0
  • B | 𝑧 + 1 | + | | 𝑧 3 2 + 1 + 3 2 𝑖 | | = 1 0
  • C | 𝑧 + 1 | + | | 𝑧 3 2 + 1 + 3 2 𝑖 | | = 6
  • D | | 𝑧 + 1 2 2 𝑖 | | + | | 𝑧 3 2 + 1 + 3 2 𝑖 | | = 6
  • E | | 𝑧 + 1 + 2 2 𝑖 | | + | | 𝑧 4 2 + 1 + 4 2 𝑖 | | = 6

Q3:

A complex number 𝑧 satisfies 2 | 𝑧 + 3 𝑖 | = 3 | 𝑧 + 1 𝑖 | . Find the Cartesian equation of the locus of 𝑧 and describe it geometrically.

  • A ( 𝑥 6 ) + 𝑦 = 2 5 , a circle with center ( 6 , 0 ) and radius 5
  • B 𝑥 3 5 + ( 𝑦 1 ) = 1 4 4 2 5 , a circle with center 3 5 , 1 and radius 1 2 5
  • C 𝑦 = 1 , the perpendicular bisector of the line segment joining ( 3 , 1 ) and ( 1 , 1 )
  • D ( 𝑥 3 ) + 𝑦 = 2 3 , a circle with center ( 3 , 0 ) and radius 2 3
  • E 𝑦 = 1 , the perpendicular bisector of the line segment joining ( 3 , 1 ) and ( 1 , 1 )

Q4:

Two loci are given by the equations | 𝑧 | = | 𝑧 + 4 𝑖 | and | 𝑧 | = 3 .

The complex numbers 𝑤 and 𝑦 satisfy both equations. Write these numbers in the form 𝑎 + 𝑖 𝑏 , where 𝑎 and 𝑏 are real numbers.

  • A 𝑤 = 5 2 𝑖 , 𝑦 = 5 2 𝑖
  • B 𝑤 = 5 2 𝑖 , 𝑦 = 5 2 𝑖
  • C 𝑤 = 5 + 2 𝑖 , 𝑦 = 5 2 𝑖
  • D 𝑤 = 2 5 𝑖 , 𝑦 = 2 + 5 𝑖
  • E 𝑤 = 5 + 2 𝑖 , 𝑦 = 5 2 𝑖

Q5:

𝑧 is a complex number such that 𝑧 + 2 3 𝑖 𝑧 4 5 𝑖 is a real number. Find the Cartesian equation of the locus of point 𝑧 and describe it geometrically.

  • A 𝑥 𝑦 4 𝑥 𝑦 + 1 = 0
  • B ( 𝑥 1 ) + ( 𝑦 4 ) = 1 0 , a circle with center ( 1 , 4 ) and radius 1 0
  • C ( 𝑥 1 ) + ( 𝑦 4 ) = 6 , a circle with center ( 1 , 4 ) and radius 6
  • D 𝑦 = 1 3 𝑥 + 1 1 3 , a straight line of slope 1 3 and 𝑦 -intercept 1 1 3
  • E 𝑦 = 1 3 𝑥 1 3 , a straight line of slope 1 3 and 𝑦 -intercept 1 3

Q6:

Consider point 𝑧 in the complex plane.

Find algebraically the locus of point 𝑧 such that | 𝑧 1 | = | 𝑧 + 𝑖 | .

  • AIt is the line of the equation 𝑦 = 𝑥 .
  • BIt is the circle of the equation ( 𝑥 + 1 ) + ( 𝑦 1 ) = 1 .
  • CIt is the circle of the equation ( 𝑥 1 ) + ( 𝑦 + 1 ) = 1 .
  • DIt is the line of the equation 𝑦 = 𝑥 .
  • EIt is the line of the equation 𝑦 = 2 𝑥 .

Which of the following is a correct geometrical description of the locus of point 𝑧 such that | 𝑧 1 | = | 𝑧 + 𝑖 | ?

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

Q7:

𝑧 is a complex number such that 𝑧 2 𝑖 + 3 𝑧 1 is an imaginary number. Find the Cartesian equation of the locus of point 𝑧 and describe it geometrically.

  • A 𝑦 = 1 2 𝑥 + 1 2 , a straight line of slope 1 2 and 𝑦 -intercept 1 2
  • B ( 𝑥 + 1 ) + ( 𝑦 + 1 ) = 5 , a circle with center ( 1 , 1 ) and radius 5
  • C 𝑦 = 1 3 𝑥 + 1 3 , a straight line of slope 1 3 and 𝑦 -intercept 1 3
  • D ( 𝑥 1 ) + ( 𝑦 + 1 ) = 5 , a circle with center ( 1 , 1 ) and radius 5
  • E ( 𝑥 + 1 ) + ( 𝑦 1 ) = 5 , a circle with center ( 1 , 1 ) and radius 5

Q8:

Describe the locus of 𝑧 such that | 𝑧 + 4 𝑖 | = 5 and give its Cartesian equation.

  • AA circle centered at 4 𝑖 with radius 5, 𝑥 + ( 𝑦 + 4 ) = 2 5
  • BA circle centered at 4 𝑖 with radius 5, 𝑥 + ( 𝑦 + 4 ) = 5
  • CA circle centered at 4 𝑖 with radius 5, 𝑥 + ( 𝑦 4 ) = 2 5
  • DA circle centered at 4 𝑖 with radius 5, 𝑥 + ( 𝑦 + 4 ) = 5
  • EA circle centered at 4 𝑖 with radius 5, 𝑥 + ( 𝑦 4 ) = 2 5

Q9:

Consider point 𝑧 in the complex plane.

Find algebraically the locus of point 𝑧 such that | 𝑧 1 3 𝑖 | = | 𝑧 3 3 𝑖 | .

  • AIt is the circle of the equation ( 𝑥 2 ) + ( 𝑦 1 ) = 9 .
  • BIt is the line of the equation 𝑥 = 2 .
  • CIt is the line of the equation 𝑦 = 3 .
  • DIt is the line of the equation 𝑦 = 3 .
  • EIt is the line of the equation 𝑥 = 2 .

Which of the following is a correct geometrical description of the locus of point 𝑧 such that | 𝑧 1 3 𝑖 | = | 𝑧 3 3 𝑖 | ?

  • AIt is the circle with center ( 2 , 1 ) and radius 3.
  • BIt is the line joining 1 + 3 𝑖 and 3 + 3 𝑖 .
  • CIt is the perpendicular bisector of the line segment joining 1 + 3 𝑖 and 3 + 3 𝑖 .
  • DIt is the perpendicular bisector of the line segment joining 1 3 𝑖 and 3 3 𝑖 .
  • EIt is the line joining 1 3 𝑖 and 3 3 𝑖 .

Q10:

Describe the locus of 𝑧 such that | 𝑧 2 | = 3 and give its Cartesian equation.

  • AA circle centered at ( 2 , 0 ) with radius 3, ( 𝑥 2 ) + 𝑦 = 3
  • BA circle centered at ( 2 , 0 ) with radius 3, ( 𝑥 2 ) + 𝑦 = 9
  • CA circle centered at ( 2 , 0 ) with radius 3, ( 𝑥 2 ) + 𝑦 = 3
  • DA circle centered at ( 3 , 0 ) with radius 2, ( 𝑥 3 ) + 𝑦 = 4
  • EA circle centered at ( 2 , 0 ) with radius 3, ( 𝑥 2 ) + 𝑦 = 9

Q11:

Describe the locus of 𝑧 such that | 𝑧 | = 4 and give its Cartesian equation.

  • AA circle centered at the origin with radius 4, 𝑥 + 𝑦 = 1 6
  • BA circle centered at the point ( 4 , 0 ) with radius 2, ( 𝑥 4 ) + 𝑦 = 2
  • CA circle centered at the origin with radius 2, 𝑥 + 𝑦 = 4
  • DA circle centered at the origin with radius 2, 𝑥 + 𝑦 = 2
  • EA circle centered at the origin with radius 4, 𝑥 + 𝑦 = 4

Q12:

Consider point 𝑧 in the complex plane.

Find algebraically the locus of point 𝑧 such that | 𝑧 | = | 𝑧 4 | .

  • AIt is the line of the equation 𝑦 = 2 .
  • BIt is the line of the equation 𝑦 = 2 .
  • CIt is the line of the equation 𝑥 = 2 .
  • DIt is the circle of the equation ( 𝑥 2 ) + 𝑦 = 1 6 .
  • EIt is the line of the equation 𝑥 = 2 .

Which of the following is a correct geometrical description of the locus of point 𝑧 such that | 𝑧 | = | 𝑧 4 | ?

  • AIt is the perpendicular bisector of the line segment joining ( 0 , 0 ) and ( 4 , 0 ) .
  • BIt is the perpendicular bisector of the line segment joining ( 0 , 0 ) and ( 4 , 0 ) .
  • CIt is the perpendicular bisector of the line segment joining ( 0 , 0 ) and ( 0 , 4 ) .
  • DIt is the circle with center ( 2 , 0 ) and radius 4.
  • EIt is the perpendicular bisector of the line segment joining ( 0 , 0 ) and ( 0 , 4 ) .

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