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 𝑦 = 𝑥 + 4 , 𝑥 < 5 . Write an equation for the half line in the form a r g ( 𝑧 𝑎 ) = 𝜃 , where 𝑎 and 𝜋 < 𝜃 𝜋 are constants to be found.

  • A a r g ( 𝑧 ( 5 9 𝑖 ) = 3 𝜋 4
  • B a r g ( 𝑧 ( 5 + 9 𝑖 ) = 𝜋 4
  • C a r g ( 𝑧 ( 5 9 𝑖 ) = 3 𝜋 4
  • D a r g ( 𝑧 ( 5 + 9 𝑖 ) = 𝜋 4
  • E a r g ( 𝑧 ( 5 + 9 𝑖 ) = 3 𝜋 4

Q2:

Find the Cartesian equation of the locus of 𝑧 such that a r g ( 𝑧 ) = 𝜋 3 .

  • A 𝑦 = 1 3 𝑥 , 𝑥 < 0
  • B 𝑦 = 3 𝑥 , 𝑥 > 0
  • C 𝑦 = 3 𝑥 , 𝑥 < 0
  • D 𝑦 = 3 𝑥 , 𝑥 > 0
  • E 𝑦 = 1 3 𝑥 , 𝑥 > 0

Q3:

Consider 𝑧 , 𝑣 , and 𝑤 in the complex plane.

Find the Cartesian equation of the locus of 𝑧 such that | 𝑧 + 1 + 4 𝑖 | = 1 2 | 𝑧 + 4 + 4 𝑖 | .

  • A 𝑥 + ( 𝑦 + 2 ) = 8
  • B ( 𝑥 2 ) + ( 𝑦 + 4 ) = 1 8
  • C 𝑥 + 𝑦 = 8
  • D 𝑥 + ( 𝑦 + 4 ) = 4
  • E 𝑥 = 1 5 6

Find the Cartesian equation of the locus of 𝑣 such that a r g ( 𝑣 ) = 𝜋 3 .

  • A 𝑦 = 3 𝑥 , 𝑥 < 0
  • B 𝑦 = 3 𝑥 , 𝑥 > 0
  • C 𝑦 = 3 𝑥 , 𝑥 < 0
  • D 𝑦 = 1 3 𝑥 , 𝑥 > 0
  • E 𝑦 = 3 𝑥 , 𝑥 > 0

Where does the locus of 𝑧 meet the locus of 𝑣 ?

  • AAt 3 3 𝑖
  • BAt 0 . 5 3 5 + 0 . 9 2 7 𝑖 and 1 . 4 0 1 2 . 4 2 7 𝑖
  • CThe two loci do not meet.
  • DAt 3 3 𝑖 and 3 3 𝑖
  • EAt 3 3 𝑖

Find the Cartesian equation of the locus of 𝑤 such that a r g ( 𝑤 + 3 𝑖 ) = 3 𝜋 4 .

  • A 𝑦 = 𝑥 3 , 𝑥 > 0
  • B 𝑦 = 𝑥 3 , 𝑥 > 0
  • C 𝑦 = 𝑥 + 3 , 𝑥 < 0
  • D 𝑦 = 𝑥 3 , 𝑥 < 0
  • E 𝑦 = 𝑥 3 , 𝑥 < 0

Where does the locus of 𝑧 meet the locus of 𝑤 ?

  • AAt 1 7 2 + 7 7 2 𝑖 and 1 + 7 2 7 + 7 2 𝑖
  • BAt 1 + 7 2 7 + 7 2 𝑖
  • CAt 1 7 2 + 7 + 7 2 𝑖 and 1 + 7 2 7 7 2 𝑖
  • DThe two loci do not meet.
  • EAt 1 7 2 + 7 7 2 𝑖

Q4:

A half line is given by 𝑦 = 𝑥 3 , 𝑥 > 2 . Write an equation for the half line in the form a r g ( 𝑧 𝑎 ) = 𝜃 , where 𝑎 and 𝜋 < 𝜃 𝜋 are constants to be found.

  • A a r g ( 𝑧 ( 3 ) ) = 𝜋 4
  • B a r g ( 𝑧 ( 3 ) ) = 𝜋 4
  • C a r g ( 𝑧 ( 2 𝑖 ) ) = 𝜋 4
  • D a r g ( 𝑧 ( 2 + 𝑖 ) ) = 𝜋 4
  • E a r g ( 𝑧 ( 2 𝑖 ) = 𝜋 4

Q5:

Find the Cartesian equation of the locus of 𝑧 such that a r g ( 𝑧 ) = 3 𝜋 4 .

  • A 𝑦 = 1 2 𝑥 , 𝑥 < 0
  • B 𝑦 = 2 𝑥 , 𝑥 < 0
  • C 𝑦 = 2 𝑥 , 𝑥 < 0
  • D 𝑦 = 𝑥 , 𝑥 < 0
  • E 𝑦 = 𝑥 , 𝑥 < 0

Q6:

Consider 𝑧 and 𝑤 in the complex plane.

Find the Cartesian equation of the locus of 𝑧 such that | 𝑧 2 3 𝑖 | = | 𝑧 + 1 + 𝑖 | .

  • A 𝑦 = 3 4 𝑥 1 1 8
  • B 𝑦 = 3 4 𝑥 + 1 1 8
  • C 𝑦 = 3 4 𝑥 1 5 8
  • D 𝑦 = 3 4 𝑥 + 1 5 8
  • E 𝑦 = 3 4 𝑥 + 1 1 8

Find the Cartesian equation of the locus of 𝑤 such that a r g ( 𝑤 + 4 2 𝑖 ) = 𝜋 4 .

  • A 𝑦 = 𝑥 6 , 𝑥 > 4
  • B 𝑦 = 𝑥 2 , 𝑥 > 4
  • C 𝑦 = 𝑥 2 , 𝑥 > 4
  • D 𝑦 = 𝑥 + 2 , 𝑥 > 4
  • E 𝑦 = 𝑥 6 , 𝑥 > 4

Find the point at which the two loci meet.

  • AThe two loci do not meet.
  • B 2 7 2 3 5 4 𝑖
  • C 2 7 2 + 2 3 2 𝑖
  • D 5 5 8 + 2 0 9 3 2 𝑖
  • E 2 7 1 4 + 7 9 2 8 𝑖

Q7:

Find the Cartesian equation of the locus of 𝑧 such that a r g ( 𝑧 ) = 5 𝜋 6 .

  • A 𝑦 = 1 3 𝑥 , 𝑥 > 0
  • B 𝑦 = 3 𝑥 , 𝑥 > 0
  • C 𝑦 = 3 𝑥 , 𝑥 > 0
  • D 𝑦 = 1 3 𝑥 , 𝑥 > 0
  • E 𝑦 = 1 3 𝑥 , 𝑥 < 0

Q8:

Which of the graphs shown is the correct representation of the locus of 𝑧 that satisfies a r g ( 𝑧 ) = 3 𝜋 4 ?

  • A(b)
  • B(a)
  • C(e)
  • D(d)
  • E(c)

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