Problem 4 - Advanced Engineering Mathematics

Question

Sketch the graph of the given equation. $$ \operatorname{Im}(z-i)=\operatorname{Re}(z+4-3 i) $$

Step-by-Step Solution

Answer

The graph is a line: \(y = x + 5\).

Step 1: Understanding the Equation

We are given the equation \(\operatorname{Im}(z-i) = \operatorname{Re}(z+4-3i)\), where \(z\) is a complex number written as \(z = x + yi\). We need to rewrite it by expressing the imaginary and real parts.

Step 2: Separate Imaginary Part

The imaginary part of \(z-i\) is \(\operatorname{Im}(z-i) = \operatorname{Im}((x + yi) - i) = y - 1\). Thus, \(\operatorname{Im}(z-i) = y - 1\).

Step 3: Separate Real Part

The real part of \(z + 4 - 3i\) is \(\operatorname{Re}(z + 4 - 3i) = \operatorname{Re}((x + yi) + 4 - 3i) = x + 4\). Thus, \(\operatorname{Re}(z + 4 - 3i) = x + 4\).

Step 4: Rewrite the Equation

Now substitute the separated parts into the equation: \(y - 1 = x + 4\).

Step 5: Simplify and Rearrange

Simplify the equation: \(y - 1 = x + 4\), which becomes \(y = x + 5\) after rearranging.

Step 6: Graph the Equation

The equation \(y = x + 5\) is the equation of a straight line with a slope of 1 and y-intercept at 5. Sketch this line on the coordinate plane, crossing the y-axis at 5 and having a slope that rises 1 unit up for every 1 unit it moves right.

Key Concepts

Imaginary and Real Parts

Complex numbers are composed of two parts: a real part and an imaginary part. This is typically represented in the form \(z = x + yi\), where \(x\) is the real part and \(yi\) is the imaginary part, with \(i\) representing the square root of \(-1\). In dealing with equations involving complex numbers, it's essential to clearly identify these parts.

In the given exercise, we start with the equation \(\operatorname{Im}(z-i) = \operatorname{Re}(z+4-3i)\). Here, \(z\) represents a complex number. Breaking it down:

• \(\operatorname{Im}(z-i)\): Remove the imaginary unit \(i\) from \(z\), effectively adjusting the imaginary part of the complex number. Thus, it is calculated by \((x + yi) - i = y - 1\).
• \(\operatorname{Re}(z+4-3i)\): This expression adjusts the real part of \(z\) by adding 4, resulting in \((x + 4)\).

In these two expressions, it's key to rearrange and identify the real and imaginary parts separately to solve or further analyze the equation.

Equation of a Line

An equation like \(y = x + 5\) is recognized as a linear equation, which is the equation of a straight line in coordinate geometry. To understand and draw this line, we identify two key characteristics: the slope and the y-intercept.

The slope is the coefficient of \(x\) in the equation, which in this case is 1. This indicates that for every step you move horizontally (in the x-direction), you move an equal amount vertically (in the y-direction). Such a slope results in a line that inclines directly 1 unit up for every 1 unit moved to the right.

The y-intercept is the constant added to the equation, seen here as 5. This tells us where the line crosses the y-axis: at the point \((0, 5)\).

• Slope: 1 (Rises 1 unit for each unit moved right)
• Y-intercept: 5 (Crosses y-axis at (0, 5))

With these two features, we can pinpoint a line's direction and position on the coordinate plane.

Coordinate Plane

The coordinate plane is a two-dimensional surface on which we plot points, lines, and shapes using algebraic equations. It is defined by horizontal and vertical intersecting lines called axes: the x-axis (horizontal) and the y-axis (vertical).

Understanding how to graph equations on the coordinate plane is essential. For our given equation \(y = x + 5\):

• Begin at the y-intercept, (0, 5), and mark this point on the y-axis.
• Utilize the slope of the equation, which guides movement from the y-intercept. From the point (0, 5), move 1 unit up and 1 unit to the right.
• Plot this new point and draw a straight line passing through both points.

This process visualizes the equation graphically, helping in understanding its behavior across the coordinate plane. A well-drawn graph reveals the relationship between variables, offering a powerful tool to visualize mathematical relationships.