Lesson Worksheet: Loci in the Complex Plane Using the Modulus Mathematics

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 |𝑧2+4𝑖|=|𝑧+2+2𝑖|.

Describe the locus of 𝑧 and give its Cartesian equation.

  • AThe perpendicular bisector of the line segment between 2+4𝑖 and 2+2𝑖, equation: 𝑦=2𝑥+3
  • BThe line joining 24𝑖 and 22𝑖, equation: 𝑦=12𝑥3
  • CThe perpendicular bisector of the line segment between 24𝑖 and 22𝑖, equation: 𝑦=2𝑥3
  • DThe circle with center (2,+4) and radius 1, equation: (𝑥+2)+(𝑦4)=1
  • EIt is the line joining 2+4𝑖 and 2+2𝑖, equation: 𝑦=12𝑥+3

What is the minimum value of |𝑧|?

  • A355
  • B5
  • C358
  • D6105
  • E853

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 𝐴𝐵. 𝑃 represents the number 1, 𝑄 the number 32132𝑖, 𝐴 the number 12+2𝑖, and 𝐵 the number 42142𝑖. Using the properties of an ellipse, write an equation for the locus of point 𝑧 in terms of 𝑧.

  • A||𝑧+122𝑖||+||𝑧32+1+32𝑖||=6
  • B|𝑧+1|+||𝑧32+1+32𝑖||=10
  • C|𝑧+1|+||𝑧32+1+32𝑖||=6
  • D||𝑧+1+22𝑖||+||𝑧42+1+42𝑖||=6
  • E||𝑧+1+22𝑖||+||𝑧42+1+42𝑖||=10

Q3:

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

  • A(𝑥6)+𝑦=25, a circle with center (6,0) and radius 5
  • B𝑦=1, the perpendicular bisector of the line segment joining (3,1) and (1,1)
  • C(𝑥3)+𝑦=23, a circle with center (3,0) and radius 23
  • D𝑦=1, the perpendicular bisector of the line segment joining (3,1) and (1,1)
  • E𝑥35+(𝑦1)=14425, a circle with center 35,1 and radius 125

Q4:

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

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

  • A𝑤=5+2𝑖,𝑦=52𝑖
  • B𝑤=25𝑖,𝑦=2+5𝑖
  • C𝑤=52𝑖,𝑦=52𝑖
  • D𝑤=5+2𝑖,𝑦=52𝑖
  • E𝑤=52𝑖,𝑦=52𝑖

Q5:

𝑧 is a complex number such that 𝑧+23𝑖𝑧45𝑖 is a real number. Find the Cartesian equation of the locus of point 𝑧 and describe it geometrically.

  • A(𝑥1)+(𝑦4)=6, a circle with center (1,4) and radius 6
  • B(𝑥1)+(𝑦4)=10, a circle with center (1,4) and radius 10
  • C𝑥𝑦4𝑥𝑦+1=0
  • D𝑦=13𝑥13, a straight line of slope 13 and 𝑦-intercept 13
  • E𝑦=13𝑥+113, a straight line of slope 13 and 𝑦-intercept 113

Q6:

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

  • AA circle centered at 4𝑖 with radius 5, 𝑥+(𝑦4)=25
  • BA circle centered at 4𝑖 with radius 5, 𝑥+(𝑦+4)=5
  • CA circle centered at 4𝑖 with radius 5, 𝑥+(𝑦+4)=5
  • DA circle centered at 4𝑖 with radius 5, 𝑥+(𝑦+4)=25
  • EA circle centered at 4𝑖 with radius 5, 𝑥+(𝑦4)=25

Q7:

Describe the locus of 𝑧 such that |𝑧2|=3 and give its Cartesian equation.

  • AA circle centered at (2,0) with radius 3, (𝑥2)+𝑦=9
  • BA circle centered at (2,0) with radius 3, (𝑥2)+𝑦=3
  • CA circle centered at (3,0) with radius 2, (𝑥3)+𝑦=4
  • DA circle centered at (2,0) with radius 3, (𝑥2)+𝑦=3
  • EA circle centered at (2,0) with radius 3, (𝑥2)+𝑦=9

Q8:

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

  • AA circle centered at the origin with radius 2, 𝑥+𝑦=2
  • BA circle centered at the origin with radius 4, 𝑥+𝑦=4
  • CA circle centered at the point (4,0) with radius 2, (𝑥4)+𝑦=2
  • DA circle centered at the origin with radius 4, 𝑥+𝑦=16
  • EA circle centered at the origin with radius 2, 𝑥+𝑦=4

Q9:

Consider point 𝑧 in the complex plane.

Find algebraically the locus of point 𝑧 such that |𝑧1|=|𝑧+𝑖|.

  • AIt is the line of the equation 𝑦=𝑥.
  • BIt is the circle of the equation (𝑥1)+(𝑦+1)=1.
  • CIt is the line of the equation 𝑦=𝑥.
  • DIt is the line of the equation 𝑦=2𝑥.
  • EIt is the circle of the equation (𝑥+1)+(𝑦1)=1.

Which of the following is a correct geometrical description of the locus of point 𝑧 such that |𝑧1|=|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.

Q10:

Consider point 𝑧 in the complex plane.

Find algebraically the locus of point 𝑧 such that |𝑧|=|𝑧4|.

  • AIt is the line of the equation 𝑥=2.
  • BIt is the line of the equation 𝑥=2.
  • CIt is the line of the equation 𝑦=2.
  • DIt is the circle of the equation (𝑥2)+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|=|𝑧4|?

  • 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 (0,4).
  • EIt is the perpendicular bisector of the line segment joining (0,0) and (4,0).

This lesson includes 13 additional questions for subscribers.

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