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=π‘₯+4, π‘₯<5. Write an equation for the half line in the form arg(π‘§βˆ’π‘Ž)=πœƒ, where π‘Žβˆˆβ„‚ and βˆ’πœ‹<πœƒβ‰€πœ‹ are constants to be found.

  • Aarg(π‘§βˆ’(βˆ’5βˆ’9𝑖)=βˆ’3πœ‹4
  • Barg(π‘§βˆ’(5+9𝑖)=βˆ’3πœ‹4
  • Carg(π‘§βˆ’(βˆ’5βˆ’9𝑖)=3πœ‹4
  • Darg(π‘§βˆ’(5+9𝑖)=βˆ’πœ‹4
  • Earg(π‘§βˆ’(5+9𝑖)=πœ‹4

Q2:

Find the Cartesian equation of the locus of 𝑧 such that arg(𝑧)=πœ‹3.

  • A𝑦=βˆ’βˆš3π‘₯, π‘₯>0
  • B𝑦=1√3π‘₯, π‘₯>0
  • C𝑦=√3π‘₯, π‘₯>0
  • D𝑦=√3π‘₯, π‘₯<0
  • E𝑦=1√3π‘₯, π‘₯<0

Q3:

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

Find the Cartesian equation of the locus of 𝑧 such that |𝑧+1+4𝑖|=12|𝑧+4+4𝑖|.

  • Aπ‘₯+(𝑦+4)=4
  • Bπ‘₯=βˆ’156
  • Cπ‘₯+𝑦=8
  • D(π‘₯βˆ’2)+(𝑦+4)=18
  • Eπ‘₯+(𝑦+2)=8

Find the Cartesian equation of the locus of 𝑣 such that arg(𝑣)=βˆ’πœ‹3.

  • A𝑦=√3π‘₯, π‘₯<0
  • B𝑦=βˆ’βˆš3π‘₯, π‘₯<0
  • C𝑦=√3π‘₯, π‘₯>0
  • D𝑦=1√3π‘₯, π‘₯>0
  • E𝑦=βˆ’βˆš3π‘₯, π‘₯>0

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

  • AThe two loci do not meet.
  • BAt βˆ’0.535+0.927𝑖 and 1.401βˆ’2.427𝑖
  • CAt √3βˆ’3𝑖
  • DAt βˆ’βˆš3βˆ’3𝑖
  • EAt √3βˆ’3𝑖 and βˆ’βˆš3βˆ’3𝑖

Find the Cartesian equation of the locus of 𝑀 such that arg(𝑀+3𝑖)=3πœ‹4.

  • A𝑦=βˆ’π‘₯βˆ’3, π‘₯>0
  • B𝑦=βˆ’π‘₯βˆ’3, π‘₯<0
  • C𝑦=π‘₯βˆ’3, π‘₯<0
  • D𝑦=π‘₯βˆ’3, π‘₯>0
  • E𝑦=βˆ’π‘₯+3, π‘₯<0

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

  • AAt 1βˆ’βˆš72+√7βˆ’72𝑖 and 1+√72βˆ’βˆš7+72𝑖
  • BAt 1βˆ’βˆš72+√7+72𝑖 and 1+√72βˆ’βˆš7βˆ’72𝑖
  • CThe two loci do not meet.
  • DAt 1+√72βˆ’βˆš7+72𝑖
  • EAt 1βˆ’βˆš72+√7βˆ’72𝑖

Q4:

A half line is given by 𝑦=π‘₯βˆ’3, π‘₯>2. Write an equation for the half line in the form arg(π‘§βˆ’π‘Ž)=πœƒ, where π‘Žβˆˆβ„‚ and βˆ’πœ‹<πœƒβ‰€πœ‹ are constants to be found.

  • Aarg(π‘§βˆ’(3))=πœ‹4
  • Barg(π‘§βˆ’(βˆ’2+𝑖))=πœ‹4
  • Carg(π‘§βˆ’(2βˆ’π‘–)=πœ‹4
  • Darg(π‘§βˆ’(2βˆ’π‘–))=βˆ’πœ‹4
  • Earg(π‘§βˆ’(3))=βˆ’πœ‹4

Q5:

Find the Cartesian equation of the locus of 𝑧 such that arg(𝑧)=3πœ‹4.

  • A𝑦=βˆ’βˆš2π‘₯, π‘₯<0
  • B𝑦=√2π‘₯, π‘₯<0
  • C𝑦=π‘₯, π‘₯<0
  • D𝑦=βˆ’π‘₯, π‘₯<0
  • E𝑦=βˆ’1√2π‘₯, π‘₯<0

Q6:

Consider 𝑧 and 𝑀 in the complex plane.

Find the Cartesian equation of the locus of 𝑧 such that |π‘§βˆ’2βˆ’3𝑖|=|𝑧+1+𝑖|.

  • A𝑦=βˆ’34π‘₯βˆ’158
  • B𝑦=34π‘₯+118
  • C𝑦=βˆ’34π‘₯+118
  • D𝑦=34π‘₯βˆ’118
  • E𝑦=βˆ’34π‘₯+158

Find the Cartesian equation of the locus of 𝑀 such that arg(𝑀+4βˆ’2𝑖)=βˆ’πœ‹4.

  • A𝑦=βˆ’π‘₯+2, π‘₯>βˆ’4
  • B𝑦=βˆ’π‘₯βˆ’2, π‘₯>βˆ’4
  • C𝑦=βˆ’π‘₯βˆ’2, π‘₯>4
  • D𝑦=βˆ’π‘₯βˆ’6, π‘₯>4
  • E𝑦=βˆ’π‘₯βˆ’6, π‘₯>βˆ’4

Find the point at which the two loci meet.

  • Aβˆ’2714+7928𝑖
  • Bβˆ’558+20932𝑖
  • Cβˆ’272βˆ’354𝑖
  • Dβˆ’272+232𝑖
  • EThe two loci do not meet.

Q7:

Find the Cartesian equation of the locus of 𝑧 such that arg(𝑧)=βˆ’5πœ‹6.

  • A𝑦=βˆ’βˆš3π‘₯, π‘₯>0
  • B𝑦=βˆ’1√3π‘₯, π‘₯>0
  • C𝑦=√3π‘₯, π‘₯>0
  • D𝑦=1√3π‘₯, π‘₯<0
  • E𝑦=1√3π‘₯, π‘₯>0

Q8:

Which of the graphs shown is the correct representation of the locus of 𝑧 that satisfies arg(𝑧)=βˆ’3πœ‹4?

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

Q9:

Which of the graphs shown is the correct representation of the locus of 𝑧 that satisfies arg(𝑧)=πœ‹6?

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

Q10:

Which of the graphs shown is the correct representation of the locus of 𝑧 that satisfies arg(𝑧+4+2𝑖)=βˆ’πœ‹5?

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

Q11:

Which of following figures is the correct representation of the locus of 𝑧 which satisfies argο€½π‘§βˆ’5+4𝑖𝑧+2βˆ’4𝑖=5πœ‹18?

  • Ae
  • Bc
  • Cd
  • Da
  • Eb

Q12:

Which of the following figures is the correct representation of the locus of 𝑧 which satisfies arg(π‘§βˆ’2+3𝑖)=πœ‹8?

  • Ac
  • Bd
  • Ca
  • De
  • Eb

Q13:

The figure shows a locus of a point 𝑧 in the complex plane. Write an equation for the locus in the form argο€»π‘§βˆ’π‘Žπ‘§βˆ’π‘ο‡=πœƒ, where π‘Ž,π‘βˆˆβ„‚ and 0<πœƒβ‰€πœ‹ are constants to be found.

  • Aargο€½π‘§βˆ’(βˆ’3+𝑖)π‘§βˆ’(4βˆ’3𝑖)=πœ‹5
  • Bargο€½π‘§βˆ’(4βˆ’3𝑖)π‘§βˆ’(βˆ’3+𝑖)=βˆ’πœ‹5
  • Cargο€½π‘§βˆ’(βˆ’3+𝑖)π‘§βˆ’(4βˆ’3𝑖)=9πœ‹5
  • Dargο€½π‘§βˆ’(4βˆ’3𝑖)π‘§βˆ’(βˆ’3+𝑖)=9πœ‹5
  • Eargο€½π‘§βˆ’(4βˆ’3𝑖)π‘§βˆ’(βˆ’3+𝑖)=πœ‹5

Q14:

Consider 𝑧 and 𝑀 in the complex plane.

Find the Cartesian equation of the locus of 𝑧 such that arg(π‘§βˆ’2βˆ’π‘–)=5πœ‹6.

  • A𝑦=βˆ’βˆš33π‘₯+1+2√33, π‘₯>2
  • B𝑦=βˆ’βˆš33π‘₯βˆ’1+2√33, π‘₯>2
  • C𝑦=√33π‘₯βˆ’1βˆ’2√33, π‘₯<2
  • D𝑦=βˆ’βˆš33π‘₯+1+2√33, π‘₯<2
  • E𝑦=βˆ’βˆš33π‘₯βˆ’1, π‘₯>2

Find the Cartesian equation of the locus of 𝑀 such that arg(𝑀+3+𝑖)=πœ‹3.

  • A𝑦=βˆ’βˆš3π‘₯βˆ’3√3+1, π‘₯>βˆ’3
  • B𝑦=√3π‘₯+3√3βˆ’1, π‘₯<βˆ’3
  • C𝑦=√3π‘₯βˆ’3√3βˆ’1, π‘₯<βˆ’3
  • D𝑦=√3π‘₯+3√3βˆ’1, π‘₯>βˆ’3
  • E𝑦=√3π‘₯+√3βˆ’1, π‘₯>βˆ’3

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

  • Aο€Ώβˆš32βˆ’74,14+5√32, πœ‹4
  • Bο€Ώβˆš32βˆ’74,βˆ’12βˆ’5√34, πœ‹4
  • Cο€Ώβˆš32+74,12βˆ’5√34, πœ‹2
  • Dο€Ώβˆš32βˆ’74,12+5√34, πœ‹2
  • Eο€Ώβˆš32βˆ’74,12+5√34, 7πœ‹6

Q15:

The point 𝑧 satisfies argο€Όπ‘§βˆ’6π‘§βˆ’6π‘–οˆ=πœ‹4. By plotting the locus on an Argand diagram, find its Cartesian equation.

  • A(𝑦+6)+(π‘₯+6)=6, 𝑦>π‘₯βˆ’6
  • B(𝑦+6)+(π‘₯+6)=6, 𝑦>6βˆ’π‘₯
  • C(π‘¦βˆ’6)+(π‘₯βˆ’6)=6, 𝑦>6βˆ’π‘₯
  • D(π‘¦βˆ’6)+(π‘₯βˆ’6)=6, 𝑦<6βˆ’π‘₯
  • E(𝑦+6)+(π‘₯+6)=6, 𝑦<6βˆ’π‘₯

Q16:

Given that 𝑧 satisfies argο€Όπ‘§βˆ’3π‘§βˆ’9=πœ‹2, by sketching the locus of 𝑧, find the range of values of Im(𝑧) and the range of values of the principle argument of 𝑧.

  • A3≀(𝑧)<9Im, 0≀(𝑧)<πœ‹2arg
  • Bβˆ’3≀(𝑧)<0Im, βˆ’πœ‹3≀(𝑧)<0arg
  • Cβˆ’6≀(𝑧)<0Im, 0≀(𝑧)<πœ‹6arg
  • Dβˆ’3≀(𝑧)<0Im, βˆ’πœ‹6≀(𝑧)<0arg
  • E0≀(𝑧)<3Im, 0≀(𝑧)<πœ‹6arg

Q17:

The figure shows a locus of a point 𝑧 in the complex plane. Write an equation for the locus in the form arg(π‘§βˆ’π‘Ž)=πœƒ, where π‘Žβˆˆβ„‚ and βˆ’πœ‹<πœƒβ‰€πœ‹ are constants to be found.

  • Aarg(π‘§βˆ’(βˆ’1+2𝑖))=βˆ’6πœ‹7
  • Barg(π‘§βˆ’(1+2𝑖))=βˆ’6πœ‹7
  • Carg(π‘§βˆ’(1βˆ’2𝑖))=βˆ’6πœ‹7
  • Darg(π‘§βˆ’(1βˆ’2𝑖))=6πœ‹7
  • Earg(π‘§βˆ’(βˆ’1+2𝑖))=6πœ‹7

Q18:

Find the Cartesian equation of the locus of 𝑧 which satisfies argο€½π‘§βˆ’3π‘–π‘§βˆ’5𝑖=2πœ‹3.

  • Aο€»π‘₯βˆ’βˆš3+(π‘¦βˆ’4)=4, π‘₯>0
  • Bο€»π‘₯+√3+(π‘¦βˆ’4)=4, π‘₯>0
  • Cο€Ώπ‘₯+√33+(π‘¦βˆ’4)=43, π‘₯>0
  • Dο€»π‘₯+√3+(π‘¦βˆ’4)=43, π‘₯>0
  • Eο€Ώπ‘₯βˆ’βˆš33+(π‘¦βˆ’4)=43, π‘₯>0

Q19:

Consider 𝑧 and 𝑀 in the complex plane.

Find the Cartesian equation of the locus of 𝑧 such that arg(𝑧)=πœ‹3.

  • A𝑦=√3π‘₯, π‘₯>0
  • B𝑦=1√3π‘₯, π‘₯>0
  • C𝑦=βˆ’βˆš3π‘₯, π‘₯>0
  • D𝑦=√3π‘₯, π‘₯<0
  • E𝑦=βˆ’βˆš3π‘₯, π‘₯<0

The locus of 𝑧 and the π‘₯-axis are tangent to a circle with radius 2 at the points 𝑃 and 𝑄 respectively. Find the coordinates of 𝑃 and 𝑄.

  • Aπ‘ƒο€»βˆš3,3, 𝑄3√3,0
  • Bπ‘ƒο€»βˆš3,3, 𝑄2√3,0
  • Cπ‘ƒο€»βˆš3,3, π‘„ο€»βˆš3,0
  • D𝑃2√3,3, π‘„ο€»βˆš3,0
  • E𝑃2√3,3, 𝑄2√3,0

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

  • Aargο€Ώπ‘€βˆ’2√3π‘€βˆ’βˆš3βˆ’3𝑖=πœ‹3
  • Bargο€Ώπ‘€βˆ’2√3π‘€βˆ’βˆš3βˆ’3𝑖=2πœ‹3
  • Cargο€Ώπ‘€βˆ’βˆš3π‘€βˆ’βˆš3βˆ’3𝑖=πœ‹3
  • Dargο€Ώπ‘€βˆ’2√3π‘€βˆ’βˆš3+3𝑖=2πœ‹3
  • Earg𝑀+2√3𝑀+√3+3𝑖=2πœ‹3

Q20:

Consider 𝑧 and 𝑀 in the complex plane.

Find the Cartesian equation of the locus of 𝑧 such that arg(π‘§βˆ’2)=βˆ’πœ‹3.

  • A𝑦=√3π‘₯βˆ’2√3, π‘₯>2
  • B𝑦=βˆ’βˆš3π‘₯+2√3, π‘₯>2
  • C𝑦=βˆ’βˆš3π‘₯βˆ’2√3, π‘₯>2
  • D𝑦=βˆ’βˆš3π‘₯βˆ’2√3, π‘₯<2
  • E𝑦=βˆ’βˆš3π‘₯+2√3, π‘₯<2

Find the Cartesian equation of the locus of 𝑀 such that arg(π‘€βˆ’1+4𝑖)=πœ‹4.

  • A𝑦=π‘₯βˆ’5, π‘₯>1
  • B𝑦=π‘₯βˆ’3, π‘₯>1
  • C𝑦=π‘₯+3, π‘₯>1
  • D𝑦=π‘₯+5, π‘₯>βˆ’1
  • E𝑦=π‘₯βˆ’5, π‘₯>βˆ’1

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

  • A√32+32+ο€Ώβˆš32+32𝑖, angle: 5πœ‹12
  • B3√32βˆ’12+ο€Ώ3√32βˆ’92𝑖, angle: 7πœ‹12
  • C3√32+12+ο€Ώ3√32βˆ’92𝑖, angle: 7πœ‹12
  • D3√32βˆ’12+ο€Ώ3√32βˆ’92𝑖, angle: 5πœ‹12
  • E√32+32+ο€Ώβˆš32βˆ’32𝑖, angle: 7πœ‹12

Q21:

Find the Cartesian equation of the locus of 𝑀 such that arg(𝑀+3+𝑖)=πœ‹3.

  • A𝑦=√3π‘₯+3√3βˆ’1, π‘₯β‰₯βˆ’3
  • B𝑦=√3π‘₯+3√3βˆ’1, π‘₯>βˆ’3
  • C𝑦=√3π‘₯βˆ’3√3+1, π‘₯β‰₯3
  • D𝑦=βˆ’βˆš3π‘₯βˆ’3√3βˆ’1, π‘₯>βˆ’3
  • E𝑦=√3π‘₯βˆ’3√3+1, π‘₯>3

Q22:

Sketch the locus of 𝑧 when arg(𝑧+2+𝑖)=πœ‹4.

  • A
  • B
  • C
  • D
  • E

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