Question

Find the equations in terms of \(x\) and \(y\) of the sets of points in the Argand diagram that satisfy the following: (a) \(\operatorname{Re} z^{2}=\operatorname{Im} z^{2}\); (b) \(\left(\operatorname{Im} z^{2}\right) / z^{2}=-i\) (c) \(\arg [z /(z-1)]=\pi / 2\).

Short answer

(a) x = y. (b) x=0 or y=0. (c) 3 different simplified processes

Step-by-step solution

Represent the complex number

Consider a complex number in the form: \(z = x + iy\).

Compute the square of the complex number

Calculate \(z^2\): \[z^2 = (x + iy)^2 = x^2 + 2ixy - y^2.\]

Identify the real and imaginary parts of \(z^2\)

From \(z^2 = x^2 + 2ixy - y^2\), identify real and imaginary parts: \(\operatorname{Re}(z^2) = x^2 - y^2\) and \(\operatorname{Im}(z^2) = 2xy\).

Part (a): Equation from the condition \(\operatorname{Re}(z^2) = \operatorname{Im}(z^2)\)

Set the real part equal to the imaginary part: \[x^2 - y^2 = 2xy.\]

Simplify the equation

Rearrange terms: \[x^2 - y^2 - 2xy = 0 \rightarrow (x - y)^2 = 0.\]

Solve for the relationship between x and y

Solve the simplified equation: \[x = y.\]

Part (b): Equation from the condition \((\operatorname{Im}(z^2))/z^2 = -i\)

Substitute \(\operatorname{Im}(z^2) = 2xy\) and \(z^2 = x^2 - y^2 + 2ixy\). The equation becomes: \[\frac{2xy}{x^2 - y^2 + 2ixy} = -i.\]

Simplify the ratio

Multiply both sides by \(x^2 - y^2 + 2ixy\): \[2xy = -i (x^2 - y^2 + 2ixy).\]

Separate real and imaginary parts

Distribute and separate parts: \[2xy = -ix^2 + iy^2 + 2xy.\]

Solve for the relationship

Equate real and imaginary parts to zero and solve: \[-ix^2 + iy^2 + 2xy = 0.\] Equate real parts: \[x^2 = y^2.\] Equate imaginary parts: \[2xy = 0.\] Then, solutions are: \[x=0 \,\text{or} \, y=0\]

Part (c): Equation from the condition \(\arg \left(\frac{z}{z-1}\right) = \frac{\pi}{2}\)

Express \(\arg \left(\frac{z}{z-1}\right) = \frac{\pi}{2}\). This implies that \(\frac{z}{z-1}\) is purely imaginary.

Simplify the fraction

Find \(\frac{z}{z-1}\): \[ \frac{x + iy}{(x-1) + iy}.\]

Multiply by the conjugate

Multiply numerator and denominator by the conjugate of the denominator: \[ \frac{(x+iy)((x-1)-iy)}{((x-1)+iy)((x-1)-iy)} = \frac{((xx-1) - xiy + iyx - i^2y^2)}{((x-1)^2)- (iy)^2}.\]

Simplify and separate parts

Simplify and separate into real and imaginary parts: \[\frac{(xx-x-yy)+i(-y)}{(x^2-2x+1)-(-y^2)}.\]

Set real part to zero

Since fraction must be purely imaginary: set real parts of the numerator to zero, and get: \[xx-x+1=yy-y(1-y).\]

Solve the equation

solve for: get two equations one from R.H.S and other by equating left hand side: \[{xx/(1-x^2 +2/3 alternative)}\].

Key concepts

Complex Number Operations

Complex numbers are expressed in the form: \( z = x + iy \). Here, \(x\) represents the real part and \(y\) represents the imaginary part. To perform operations on complex numbers, such as addition, subtraction, multiplication, and division, treat \(i\) as the imaginary unit, where \(i^2 = -1\).

For instance, consider a problem where you need to find \(z^2\). Let \(z = x + iy\). Squaring it, we obtain:
\[ z^2 = (x + iy)^2 = x^2 + 2ixy - y^2. \]
Here, \(x^2 - y^2\) is the real part and \(2ixy\) is the imaginary part.

The steps usually include identifying these parts separately to solve problems involving conditions placed on complex numbers, like the ones given in the problem statement. Remember, dealing with complex number operations often requires breaking down the number into its real and imaginary components and then re-combining them after performing the required algebraic operations.

Real and Imaginary Parts

Understanding the real and imaginary parts of complex numbers is crucial for various calculations. Any complex number \(z\) can be written as \(z = x + iy\), where \(x\) and \(y\) represent the real and imaginary parts respectively.

For example, in the exercise, we need to find the real and imaginary parts of \(z^2\). Starting with \(z = x + iy\), we compute:
\[ z^2 = (x + iy)^2 = x^2 + 2ixy - y^2. \]
Here,
\begin{itemize}

The real part is \(\text{Re}(z^2) = x^2 - y^2\)

The imaginary part is \(\text{Im}(z^2) = 2xy\)

Recognizing and separating these parts is fundamental when given conditions involving these components.
Consider the condition \(\text{Re}(z^2) = \text{Im}(z^2)\). This translates algebraically to:
\[ x^2 - y^2 = 2xy \]
By simplifying, we find:
\[ (x - y)^2 = 0 \rightarrow x = y. \]
Such equations help us analyze the behavior of complex numbers under given constraints.

Argand Plane Transformations

The Argand plane (or Argand diagram) is a graphical representation of complex numbers, where the x-axis represents the real part and the y-axis represents the imaginary part of the complex number. Transformations on the Argand plane involve modifying the complex number in specific ways to satisfy certain conditions.

For example, consider the condition \(\text{arg}\frac{z}{z-1}=\frac{\theta}{2}\):
\begin{itemize}

Representing \(z = x + iy\).

Finding \(\frac{z}{z-1}\), which involves dividing \(z\) by \(z-1\).

Transforming this division into its simplified form while maintaining the real and imaginary parts.

Equating the appropriate parts to fit the condition of the argument being \(\frac{\theta}{2}\).

The transformation provides a clearer visual understanding. For the example in the exercise:
Given \(\text{arg}\frac{z}{z-1}=\frac{\theta}{2}\), we initially express \(\frac{z}{z-1}\) as:
\[ \frac{x + iy}{(x-1) + iy}\]
Multiplying by the conjugate of the denominator:
\[ \frac{(x + iy) \times (x-1 - iy)}{(x-1 + iy) \times (x-1 - iy)}. \]
Simplifying further, we must set the real part of the result to zero, confirming it is purely imaginary, as required by the argument condition. This illustrates a specific transformation in the Argand plane to achieve the desired condition or restriction.