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 |π‘§βˆ’2+4𝑖|=|𝑧+2+2𝑖|.

Describe the locus of 𝑧 and give its Cartesian equation.

  • AThe perpendicular bisector of the line segment between βˆ’2+4𝑖 and 2+2𝑖, equation: 𝑦=2π‘₯+3
  • BThe line joining 2βˆ’4𝑖 and βˆ’2βˆ’2𝑖, equation: 𝑦=βˆ’12π‘₯βˆ’3
  • CThe perpendicular bisector of the line segment between 2βˆ’4𝑖 and βˆ’2βˆ’2𝑖, equation: 𝑦=2π‘₯βˆ’3
  • DThe circle with center (βˆ’2,+4) and radius 1, equation: (π‘₯+2)+(π‘¦βˆ’4)=1
  • EIt is the line joining βˆ’2+4𝑖 and 2+2𝑖, equation: 𝑦=βˆ’12π‘₯+3

What is the minimum value of |𝑧|?

  • A3√55
  • B√5
  • C3√58
  • D6√105
  • E8√53

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 𝐴𝐡. 𝑃 represents the number βˆ’1, 𝑄 the number 3√2βˆ’1βˆ’3√2𝑖, 𝐴 the number βˆ’1βˆ’βˆš2+√2𝑖, and 𝐡 the number 4√2βˆ’1βˆ’4√2𝑖. Using the properties of an ellipse, write an equation for the locus of point 𝑧 in terms of 𝑧.

  • A||𝑧+1√2βˆ’βˆš2𝑖||+||π‘§βˆ’3√2+1+3√2𝑖||=6
  • B|𝑧+1|+||π‘§βˆ’3√2+1+3√2𝑖||=10
  • C|𝑧+1|+||π‘§βˆ’3√2+1+3√2𝑖||=6
  • D||𝑧+1+√2βˆ’βˆš2𝑖||+||π‘§βˆ’4√2+1+4√2𝑖||=6
  • E||𝑧+1+√2βˆ’βˆš2𝑖||+||π‘§βˆ’4√2+1+4√2𝑖||=10

Q3:

A complex number 𝑧 satisfies 2|𝑧+3βˆ’π‘–|=3|𝑧+1βˆ’π‘–|. Find the Cartesian equation of the locus of 𝑧 and describe it geometrically.

  • A(π‘₯βˆ’6)+𝑦=25, a circle with center (6,0) and radius 5
  • B𝑦=βˆ’1, the perpendicular bisector of the line segment joining (3,βˆ’1) and (1,βˆ’1)
  • C(π‘₯βˆ’3)+𝑦=23, a circle with center (3,0) and radius √23
  • D𝑦=1, the perpendicular bisector of the line segment joining (βˆ’3,1) and (βˆ’1,1)
  • Eο€Όπ‘₯βˆ’35+(π‘¦βˆ’1)=14425, a circle with center ο€Ό35,1 and radius 125

Q4:

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

The complex numbers 𝑀 and 𝑦 satisfy both equations. Write these numbers in the form π‘Ž+𝑖𝑏, where π‘Ž and 𝑏 are real numbers.

  • A𝑀=βˆ’βˆš5+2𝑖,𝑦=βˆ’βˆš5βˆ’2𝑖
  • B𝑀=2βˆ’βˆš5𝑖,𝑦=2+√5𝑖
  • C𝑀=βˆ’βˆš5βˆ’2𝑖,𝑦=√5βˆ’2𝑖
  • D𝑀=√5+2𝑖,𝑦=√5βˆ’2𝑖
  • E𝑀=βˆ’βˆš5βˆ’2𝑖,𝑦=√5βˆ’2𝑖

Q5:

𝑧 is a complex number such that 𝑧+2βˆ’3π‘–π‘§βˆ’4βˆ’5𝑖 is a real number. Find the Cartesian equation of the locus of point 𝑧 and describe it geometrically.

  • A(π‘₯βˆ’1)+(π‘¦βˆ’4)=6, a circle with center (1,4) and radius √6
  • B(π‘₯βˆ’1)+(π‘¦βˆ’4)=10, a circle with center (1,4) and radius √10
  • Cπ‘₯π‘¦βˆ’4π‘₯βˆ’π‘¦+1=0
  • D𝑦=13π‘₯βˆ’13, a straight line of slope 13 and 𝑦-intercept βˆ’13
  • E𝑦=13π‘₯+113, a straight line of slope 13 and 𝑦-intercept 113

Q6:

Consider point 𝑧 in the complex plane.

Find algebraically the locus of point 𝑧 such that |π‘§βˆ’1|=|𝑧+𝑖|.

  • AIt is the line of the equation 𝑦=βˆ’π‘₯.
  • BIt is the circle of the equation (π‘₯βˆ’1)+(𝑦+1)=1.
  • CIt is the line of the equation 𝑦=π‘₯.
  • DIt is the line of the equation 𝑦=2π‘₯.
  • EIt is the circle of the equation (π‘₯+1)+(π‘¦βˆ’1)=1.

Which of the following is a correct geometrical description of the locus of point 𝑧 such that |π‘§βˆ’1|=|𝑧+𝑖|?

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

Q7:

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

  • A(π‘₯+1)+(π‘¦βˆ’1)=5, a circle with center (βˆ’1,1) and radius √5
  • B(π‘₯+1)+(𝑦+1)=5, a circle with center (βˆ’1,βˆ’1) and radius √5
  • C𝑦=βˆ’13π‘₯+13, a straight line of slope βˆ’13 and 𝑦-intercept 13
  • D𝑦=βˆ’12π‘₯+12, a straight line of slope βˆ’12 and 𝑦-intercept 12
  • E(π‘₯βˆ’1)+(𝑦+1)=5, a circle with center (1,βˆ’1) and radius √5

Q8:

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

  • AA circle centered at βˆ’4𝑖 with radius 5, π‘₯+(π‘¦βˆ’4)=25
  • BA circle centered at 4𝑖 with radius 5, π‘₯+(𝑦+4)=5
  • CA circle centered at βˆ’4𝑖 with radius 5, π‘₯+(𝑦+4)=5
  • DA circle centered at βˆ’4𝑖 with radius 5, π‘₯+(𝑦+4)=25
  • EA circle centered at 4𝑖 with radius 5, π‘₯+(π‘¦βˆ’4)=25

Q9:

Consider point 𝑧 in the complex plane.

Find algebraically the locus of point 𝑧 such that |π‘§βˆ’1βˆ’3𝑖|=|π‘§βˆ’3βˆ’3𝑖|.

  • AIt is the circle of the equation (π‘₯βˆ’2)+(π‘¦βˆ’1)=9.
  • BIt is the line of the equation π‘₯=βˆ’2.
  • CIt is the line of the equation 𝑦=βˆ’3.
  • DIt is the line of the equation π‘₯=2.
  • EIt is the line of the equation 𝑦=3.

Which of the following is a correct geometrical description of the locus of point 𝑧 such that |π‘§βˆ’1βˆ’3𝑖|=|π‘§βˆ’3βˆ’3𝑖|?

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

Q10:

Describe the locus of 𝑧 such that |π‘§βˆ’2|=3 and give its Cartesian equation.

  • AA circle centered at (βˆ’2,0) with radius 3, (π‘₯βˆ’2)+𝑦=9
  • BA circle centered at (βˆ’2,0) with radius 3, (π‘₯βˆ’2)+𝑦=3
  • CA circle centered at (3,0) with radius 2, (π‘₯βˆ’3)+𝑦=4
  • DA circle centered at (2,0) with radius 3, (π‘₯βˆ’2)+𝑦=3
  • EA circle centered at (2,0) with radius 3, (π‘₯βˆ’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 2, π‘₯+𝑦=2
  • BA circle centered at the origin with radius 4, π‘₯+𝑦=4
  • CA circle centered at the point (4,0) with radius 2, (π‘₯βˆ’4)+𝑦=2
  • DA circle centered at the origin with radius 4, π‘₯+𝑦=16
  • EA circle centered at the origin with radius 2, π‘₯+𝑦=4

Q12:

Consider point 𝑧 in the complex plane.

Find algebraically the locus of point 𝑧 such that |𝑧|=|π‘§βˆ’4|.

  • AIt is the line of the equation π‘₯=2.
  • BIt is the line of the equation π‘₯=βˆ’2.
  • CIt is the line of the equation 𝑦=βˆ’2.
  • DIt is the circle of the equation (π‘₯βˆ’2)+𝑦=16.
  • EIt is the line of the equation 𝑦=2.

Which of the following is a correct geometrical description of the locus of point 𝑧 such that |𝑧|=|π‘§βˆ’4|?

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

Q13:

A complex number 𝑧 satisfies |𝑧+3+3𝑖|=3.

Describe the locus of 𝑧 and give its Cartesian equation.

  • AA circle centered at 3+3𝑖 with radius 3, (π‘₯βˆ’3)+(π‘¦βˆ’3)=9
  • BA circle centered at βˆ’3βˆ’3𝑖 with radius 3, (π‘₯βˆ’3)+(π‘¦βˆ’3)=9
  • CA circle centered at 3+3𝑖 with radius 3, (π‘₯βˆ’3)+(π‘¦βˆ’3)=3
  • DA circle centered at βˆ’3βˆ’3𝑖 with radius 3, (π‘₯+3)+(𝑦+3)=9
  • EA circle centered at 3+3𝑖 with radius 3, (π‘₯+3)+(𝑦+3)=9

Find the range of the argument of 𝑧 in the range [βˆ’πœ‹,πœ‹].

  • Aο“πœ‹2,πœ‹οŸ
  • Bο“βˆ’πœ‹,βˆ’πœ‹2
  • C0,πœ‹2
  • Dο”βˆ’πœ‹4,βˆ’3πœ‹4
  • Eο“βˆ’πœ‹2,0

Find the range of the modulus of 𝑧.

  • A3√2βˆ’3,3√2+3
  • B0,3√2+3
  • C[0,3]
  • D[0,6]
  • E9βˆ’3√2,9+3√2

Q14:

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

  • A𝑦=βˆ’12, a horizontal line with constant 𝑦=βˆ’0.5
  • Bο€Όπ‘₯+43+(𝑦)=49, a circle with center ο€Όβˆ’43,0 and radius 23
  • Cπ‘₯+(𝑦+2)=2, a circle with center (0,βˆ’2) and radius √2
  • D(π‘₯)+(𝑦)=49, a circle with center (0,0) and radius 23
  • Eπ‘₯+𝑦+43=49, a circle with center ο€Ό0,βˆ’43 and radius 23

Q15:

A complex number 𝑧 satisfies |𝑧|+|π‘§βˆ’5βˆ’3𝑖|=8. Describe the locus of 𝑧.

  • AIt is an ellipse with foci at the origin and at 5+3𝑖 and has a major axis of 4.
  • BIt is an ellipse with foci at the origin and at βˆ’5βˆ’3𝑖 and has a major axis of 4.
  • CIt is an ellipse with foci at the origin and at 5+3𝑖 and has a major axis of 8.
  • DIt is an ellipse with foci at the origin and at βˆ’5βˆ’3𝑖 and has a major axis of 8.
  • EIt is a line with equation 𝑦=βˆ’53π‘₯βˆ’5.

Q16:

A complex number 𝑀 lies at a distance of 5√2 from 𝑧=92+72π‘–οŠ§ and a distance of 4√5 from 𝑧=βˆ’92βˆ’72π‘–οŠ¨. Does the point 𝑀 lie on the circle centered at the origin that passes through π‘§οŠ§ and π‘§οŠ¨?

  • ANo
  • BYes

Q17:

The figure shows a locus of a point 𝑧 in the complex plane. Write an equation for the locus in the form |𝑧+2+7𝑖|=|π‘§βˆ’π‘Ž|, where π‘Žβˆˆβ„‚ is a constant to be found.

  • A|𝑧+2+7𝑖|=|π‘§βˆ’(βˆ’6.32βˆ’20.76𝑖)|
  • B|𝑧+2+7𝑖|=|π‘§βˆ’(6.32+20.76𝑖)|
  • C|𝑧+2+7𝑖|=|π‘§βˆ’(βˆ’1.5βˆ’2𝑖)|
  • D|𝑧+2+7𝑖|=|π‘§βˆ’(2+15𝑖)|
  • E|𝑧+2+7𝑖|=|π‘§βˆ’(βˆ’10βˆ’π‘–)|

Q18:

The figure shows a circular locus of a point 𝑧 in the complex plane. Given that 𝐢 is the centre of the circle, write an equation for the locus in the form |π‘§βˆ’π‘Ž|=𝑏, where π‘Žβˆˆβ„‚ and 𝑏>0 are constants to be found.

  • A|π‘§βˆ’(βˆ’10+4𝑖)|=√29
  • B|π‘§βˆ’(βˆ’10+4𝑖)|=√21
  • C|π‘§βˆ’(βˆ’5+2𝑖)|=√21
  • D|π‘§βˆ’(5βˆ’2𝑖)|=√21
  • E|π‘§βˆ’(βˆ’5+2𝑖)|=√29

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