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AoPS - Lesson 6, Complex Plane

Critical Thinking Framework Remind yourself of the problem solving strategy/approach recommendations before you tackle the problems.

Aditional Resources:

Problem 1

Problem Statement:

The point \( \overline{z} \) is shown below. What is \( z \)?

It includes a diagram where the point \( \overline{z} \) is located at (-2, 1) on the complex plane. The task is to determine the value of the complex number \( z \), given its conjugate \( \overline{z} \).

Problem Deconstruction

The problem involves finding the complex number \( z \) when given its conjugate \( \overline{z} \). The given point \( \overline{z} \) is located at the coordinates (-2, 1) on the complex plane.

Givens

  • The point \( \overline{z} \) is located at (-2, 1).
  • In complex numbers, if \( z = a + bi \), then its conjugate \( \overline{z} = a - bi \).

Unknowns

  • The values of \( a \) and \( b \) in the expression \( z = a + bi \).
  • The value of \( z \) itself.

Constraints

  • The solution must involve only elementary methods appropriate for Algebra 1 concepts, focusing on the relationship between a complex number and its conjugate.

Hints for Strategy to Solve

  1. Identify Coordinates: Recognize that the coordinates of \( \overline{z} \) correspond to the real and imaginary parts of the complex number.
  2. Understand Conjugate Relationship: Use the definition of the complex conjugate to relate \( z \) and \( \overline{z} \).
  3. Formulate Equations: Set up equations based on the real and imaginary components derived from the coordinates of \( \overline{z} \).
  4. Solve for Unknowns: Use the equations to find the values of \( a \) and \( b \) that will give you \( z \).

Problem 2

Problem Statement:

Let \( z \) be a complex number satisfying the equation:

\[ 2z + 3\overline{z} = - 25 - 2i. \]

What is the value of \( z \)?

Deconstruction:

The equation involves both the complex number \( z \) and its conjugate \( \overline{z} \). The goal is to find the value of \( z \), given the relationship between \( z \), \( \overline{z} \), and the complex number on the right-hand side of the equation.

Givens:

  • \( z \) is a complex number, represented as \( z = a + bi \), where \( a \) is the real part and \( b \) is the imaginary part.
  • The conjugate of \( z \), denoted \( \overline{z} \), is \( \overline{z} = a - bi \).
  • The equation is \( 2z + 3\overline{z} = -25 - 2i \).

Unknowns:

  • The real part \( a \) and the imaginary part \( b \) of the complex number \( z = a + bi \).
  • The exact value of \( z \).

Constraints:

  • Use only elementary methods involving Algebra 1 concepts and the relationship between a complex number and its conjugate.

Hints for Strategy to Solve:

  1. Substitute \( z \) and \( \overline{z} \): Write \( z \) as \( a + bi \) and \( \overline{z} \) as \( a - bi \), then substitute them into the given equation.
  2. Separate Real and Imaginary Parts: Simplify the resulting expression, isolating the real and imaginary components.
  3. Form a System of Equations: This will yield two separate equations—one for the real part and one for the imaginary part. Solve this system to find \( a \) and \( b \).
  4. Solve for \( z \): Once \( a \) and \( b \) are known, you can express the value of \( z \) as \( a + bi \).

Problem 3

Problem Statement:

Find the area of the region in the complex plane defined by \( |z| < 5 \), where \( z \) is a complex number.

Deconstruction:

The expression \( |z| < 5 \) represents a region in the complex plane. The magnitude \( |z| \) of a complex number \( z = a + bi \) is given by \( |z| = \sqrt{a^2 + b^2} \). This inequality defines a circular region in the complex plane, where the distance from the origin (0, 0) to any point \( z \) is less than 5.

Givens:

  • \( |z| < 5 \) defines the set of all points inside a circle with radius 5, centered at the origin of the complex plane.
  • The region in question is a circle with radius 5.

Unknowns:

  • The area of the region inside the circle defined by \( |z| < 5 \).

Constraints:

  • The solution must involve basic geometric concepts related to the area of a circle.

Hints for Strategy to Solve:

  1. Interpret the Magnitude Condition: Recognize that \( |z| < 5 \) corresponds to a circle in the complex plane with radius 5.
  2. Use the Formula for the Area of a Circle: The area \( A \) of a circle with radius \( r \) is given by \( A = \pi r^2 \).
  3. Substitute the Radius: Plug the value \( r = 5 \) into the area formula to find the area of the region.

Problem 4

Problem Statement:

Find a function \( f(z) \) such that the line shown below is the graph of the equation \( |z - 3i| = |f(z)| \).

Deconstruction:

The problem requires finding a function \( f(z) \) that describes the relationship between the distance from the point \( z \) to \( 3i \) (the point on the imaginary axis at \( (0, 3) \)) and the distance from \( z \) to the graph of the red line, which has the equation \( y = \frac{1}{2}x + \frac{1}{2} \).

Givens:

  • The equation \( |z - 3i| = |f(z)| \) involves the distance from a point \( z \) on the complex plane to the point \( 3i \).
  • The red line represents the geometric locus where the distance from \( z \) to \( 3i \) equals the distance to some function \( f(z) \).
  • The red line is given by the equation \( y = \frac{1}{2}x + \frac{1}{2} \), which can be interpreted geometrically in the complex plane.

Unknowns:

  • The function \( f(z) \) that describes the distance relationship represented by the red line.

Constraints:

  • The function \( f(z) \) must be found such that the relationship \( |z - 3i| = |f(z)| \) holds for all points on the red line.

Hints for Strategy to Solve:

  1. Interpret \( z \) Geometrically: Let \( z = x + yi \), where \( x \) and \( y \) are the real and imaginary components of a complex number.
  2. Understand the Distance Condition: The equation \( |z - 3i| = |f(z)| \) means the distance from \( z \) to \( 3i \) is equal to the distance to some point on the red line.
  3. Parameterize the Line: The equation of the red line is \( y = \frac{1}{2}x + \frac{1}{2} \), so for any point \( z = x + yi \), the point on the red line is related to the function.
  4. Derive the Function: Analyze the geometry of the situation to deduce the correct form of \( f(z) \). Consider the relationship between the distances involved and use this to formulate \( f(z) \).

Problem 5

Problem Statement:

Evaluate the magnitude of the complex number \( \frac{1}{2} - \frac{3}{8}i \).

Deconstruction:

The task requires calculating the magnitude (or modulus) of a given complex number. The magnitude \( |z| \) of a complex number \( z = a + bi \) is found using the formula \( |z| = \sqrt{a^2 + b^2} \), where \( a \) is the real part and \( b \) is the imaginary part.

Givens:

  • The complex number is \( \frac{1}{2} - \frac{3}{8}i \), where:
  • Real part \( a = \frac{1}{2} \)
  • Imaginary part \( b = -\frac{3}{8} \)

Unknowns:

  • The magnitude \( |z| \) of the complex number.

Constraints:

  • The magnitude must be computed using elementary methods and the standard modulus formula for complex numbers.

Strategy Hints (Flow):

  1. Recognize the Complex Number Form: Identify that the given complex number is \( \frac{1}{2} - \frac{3}{8}i \), with \( a = \frac{1}{2} \) and \( b = -\frac{3}{8} \).
  2. Apply the Magnitude Formula: Use the formula \( |z| = \sqrt{a^2 + b^2} \), where \( a \) and \( b \) are the real and imaginary parts of the complex number.
  3. Square the Real and Imaginary Parts: Compute \( \left( \frac{1}{2} \right)^2 \) and \( \left( -\frac{3}{8} \right)^2 \).
  4. Add the Squares: Sum the results from squaring the real and imaginary components.
  5. Simplify and Take Square Root: Simplify the sum of squares and take the square root to find the magnitude of the complex number.

Problem 6

Problem Statement:

Find the product of the magnitudes of the complex numbers \( 3 - 2i \) and \( 3 + 2i \), expressed as \( |3 - 2i| \cdot |3 + 2i| \).

Deconstruction:

The task requires calculating the magnitudes of two complex numbers and then multiplying these magnitudes together. The magnitude (or modulus) of a complex number \( z = a + bi \) is given by the formula: [ |z| = \sqrt{a^2 + b^2} ]

Givens:

  • The complex numbers are \( 3 - 2i \) and \( 3 + 2i \).
  • For \( 3 - 2i \):
  • Real part \( a = 3 \)
  • Imaginary part \( b = -2 \)
  • For \( 3 + 2i \):
  • Real part \( a = 3 \)
  • Imaginary part \( b = 2 \)

Unknowns:

  • The magnitudes \( |3 - 2i| \) and \( |3 + 2i| \).

Constraints:

  • Use the standard formula for the modulus of complex numbers to find the magnitudes.

Strategy Hints:

  1. Calculate the Magnitude of \( 3 - 2i \):
  2. Identify the real part \( a = 3 \) and the imaginary part \( b = -2 \).

  3. Use the formula \( |3 - 2i| = \sqrt{3^2 + (-2)^2} \).

  4. Calculate the Magnitude of \( 3 + 2i \):

  5. Identify the real part \( a = 3 \) and the imaginary part \( b = 2 \).

  6. Use the formula \( |3 + 2i| = \sqrt{3^2 + 2^2} \).

  7. Multiply the Magnitudes:

  8. Use the results from the previous calculations to find the product: [ |3 - 2i| \cdot |3 + 2i| = \sqrt{13} \cdot \sqrt{13} = 13. ]

Final Result:

Thus, the value of \( |3 - 2i| \cdot |3 + 2i| \) is: [ \boxed{13} ]

Problem 7

Problem Statement:

Calculate the magnitude \( |(1 - i)^8| \).

Deconstruction:

The problem requires finding the magnitude of the complex number \( (1 - i) \) raised to the power of 8. The magnitude of a complex number can be calculated using the property that \( |z^n| = |z|^n \).

Givens:

  • The complex number is \( 1 - i \).

Unknowns:

  • The magnitude \( |(1 - i)^8| \).

Constraints:

  • Use the properties of magnitudes and complex numbers to find the result.

Strategy Hints (Flow):

  1. Calculate the Magnitude of \( 1 - i \):

  2. Identify the real part \( a = 1 \) and the imaginary part \( b = -1 \).

  3. Use the formula \( |1 - i| = \sqrt{1^2 + (-1)^2} \).

  4. Raise the Magnitude to the 8th Power:

  5. Use the property \( |(1 - i)^8| = |1 - i|^8 \) to find the final result.

Detailed Calculation Steps:

  1. Magnitude Calculation:
\[ |1 - i| = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]
  1. Raise to the 8th Power:
\[ |(1 - i)^8| = |1 - i|^8 = (\sqrt{2})^8 \]
  1. Simplify the Result:
\[ (\sqrt{2})^8 = (2^{1/2})^8 = 2^{4} = 16 \]

Final Result:

Thus, the value that Ringo should have found is:

\[ \boxed{16} \]

Problem 8

Problem Statement:

Given a complex number \( z \) that lies on the circle centered at 0 with radius 5, calculate \( |z^3| \).

Deconstruction:

The task involves finding the magnitude of \( z^3 \) based on the given information about the complex number \( z \).

Givens:

  • The radius of the circle is 5, so \( |z| = 5 \).

Unknowns:

  • The magnitude \( |z^3| \).

Constraints:

  • Use the properties of magnitudes in complex numbers.

Strategy Hints:

  1. Recall Magnitude Property: Use the property that \( |z^n| = |z|^n \) for any complex number \( z \).

  2. Identify Given Magnitude: Note that the radius of the circle indicates \( |z| = 5 \).

  3. Calculate \( |z^3| \): Apply the magnitude property to find \( |z^3| = |z|^3 \).

  4. Substitute Known Value: Substitute \( |z| = 5 \) into the equation to get the final result.

Problem 9

Problem Statement:

Given that \( z \) is a complex number such that \( z^3 = 100 + 75i \), find the magnitude \( |z| \).

Deconstruction:

The task involves finding the magnitude of the complex number \( z \) based on the magnitude of \( z^3 \).

Givens:

  • The complex number \( z^3 = 100 + 75i \).

Unknowns:

  • The magnitude \( |z| \).

Constraints:

  • Use properties of magnitudes in complex numbers to relate \( |z| \) and \( |z^3| \).

Strategy Hints:

  1. Calculate Magnitude of \( z^3 \): Use the formula for the magnitude of a complex number:
\[ |z^3| = \sqrt{(100)^2 + (75)^2} \]

  1. Apply Magnitude Property: Recall that \( |z^n| = |z|^n \) for any complex number \( z \).

  2. Relate \( |z| \) to \( |z^3| \): Since \( z^3 = 100 + 75i \), use the property to express \( |z| \):

\[ |z^3| = |z|^3 \]
  1. Solve for \( |z| \): Rearrange the equation to isolate \( |z| \) based on the magnitude of \( z^3 \).

Problem 10

Problem Statement:

Given that the square of the non-real complex number \( z \) is equal to its conjugate \( \overline{z} \), find the real part of \( z \).

Deconstruction:

The problem requires analyzing the relationship between a complex number and its conjugate, as well as finding the real part from a given equation.

Givens:

  • \( z \) is a non-real complex number.
  • The equation is \( z^2 = \overline{z} \).

Unknowns:

  • The real part of the complex number \( z \), denoted as \( \text{Re}(z) \).

Constraints:

  • Use properties of complex numbers and their conjugates to solve the equation.

Strategy Hints:

  1. Express \( z \) in Terms of Real and Imaginary Parts: Write \( z \) as \( z = x + yi \), where \( x \) is the real part and \( y \) is the imaginary part.

  2. Substitute into the Equation: Substitute \( z \) into the equation \( z^2 = \overline{z} \):

\[ (x + yi)^2 = x - yi \]

  1. Expand and Simplify: Expand the left side and equate the real and imaginary parts to form a system of equations.

  2. Analyze the Results: Use the system of equations to solve for \( x \) (the real part) and \( y \) (the imaginary part), noting that \( z \) is non-real implies \( y \neq 0 \).

  3. Extract the Real Part: Determine the value of \( x \) based on the derived equations.

Problem 11

Problem Statement:

Find all complex numbers \( z \) such that \( |z - 3| = |z + i| = |z - 3i| \).

Deconstruction:

The problem involves finding complex numbers that are equidistant from three points in the complex plane, namely \( 3 \), \( -i \), and \( 3i \).

Givens:

  • The distances are defined as:

  • \( |z - 3| \): distance from \( z \) to the point \( 3 \) (or \( 3 + 0i \)).

  • \( |z + i| \): distance from \( z \) to the point \( -i \) (or \( 0 - i \)).
  • \( |z - 3i| \): distance from \( z \) to the point \( 3i \) (or \( 0 + 3i \)).

Unknowns:

  • The complex number \( z \) that satisfies the given distance equations.

Constraints:

  • The solution must satisfy the conditions for distances in the complex plane.

Strategy Hints:

  1. Express \( z \) in Terms of Real and Imaginary Parts: Let \( z = x + yi \), where \( x \) and \( y \) are real numbers.

  2. Set Up Distance Equations: Write the equations for the distances:

  3. \( |z - 3| = |z + i| \)

  4. \( |z + i| = |z - 3i| \)

  5. Use the Distance Formula: Apply the distance formula for complex numbers:

\[ |z - a| = \sqrt{(x - \text{Re}(a))^2 + (y - \text{Im}(a))^2} \]

  1. Equate Distances: From the distance equations, set up systems of equations to relate \( x \) and \( y \).

  2. Geometric Interpretation: Consider the geometric implications of equidistance, potentially leading to finding intersections of perpendicular bisectors of the segments connecting the points \( 3 \), \( -i \), and \( 3i \).

  3. Solve the Resulting System: Solve the resulting equations to find the values of \( x \) and \( y \) that satisfy all conditions.

Problem 12

Problem Statement:

Let \( p(x) = x^3 - 5x^2 + 12x - 19 \) have roots \( a \), \( b \), and \( c \). Find the value of

\[ \frac{1}{ab} + \frac{1}{bc} + \frac{1}{ca}. \]

Deconstruction:

The task is to calculate a specific expression involving the products of the roots of a cubic polynomial.

Givens:

  • The polynomial is \( p(x) = x^3 - 5x^2 + 12x - 19 \).
  • The roots of the polynomial are \( a, b, c \).

Unknowns:

  • The value of the expression \( \frac{1}{ab} + \frac{1}{bc} + \frac{1}{ca} \).

Constraints:

  • Use Vieta's formulas, which relate the coefficients of a polynomial to sums and products of its roots.

Strategy Hints:

  1. Apply Vieta's Formulas: Recall Vieta's formulas for the roots of a cubic polynomial:
  2. \( a + b + c = 5 \) (coefficient of \( x^2 \) with a negative sign),
  3. \( ab + ac + bc = 12 \) (coefficient of \( x \)),

  4. \( abc = 19 \) (constant term with a negative sign).

  5. Rewrite the Expression: Notice that:

\[ \frac{1}{ab} + \frac{1}{bc} + \frac{1}{ca} = \frac{c + a + b}{abc}. \]

  1. Substitute Known Values: Use the values obtained from Vieta’s formulas to replace \( c + a + b \) and \( abc \).

  2. Calculate the Result: Simplify the expression to find the final value.

Solutions

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