Which diagram best represents the solutions to the equation
arg(z) = arg(z + 1 - i)?
A - HSC - SSCE Mathematics Extension 2 - Question 7 - 2018 - Paper 1
Question 7
Which diagram best represents the solutions to the equation
arg(z) = arg(z + 1 - i)?
A.
B.
C.
D.
Worked Solution & Example Answer:Which diagram best represents the solutions to the equation
arg(z) = arg(z + 1 - i)?
A - HSC - SSCE Mathematics Extension 2 - Question 7 - 2018 - Paper 1
Step 1
Identify the argument of the complex number
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Answer
To solve the equation arg(z) = arg(z + 1 - i), we must understand what each term means. The argument of a complex number z can be described by the angle it makes with the positive x-axis in a complex plane. Here, z is represented as z = x + yi.
Step 2
Translate arg(z + 1 - i)
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Answer
The expression z + 1 - i translates to (x + 1) + (y - 1)i. The argument of this new complex number is calculated similarly, forming an angle based on its x and y components.
Step 3
Determine the geometric representation
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Answer
The equation arg(z) = arg(z + 1 - i) implies that the angle defined by the line connecting the origin (0,0) to the point (x,y) must equal the angle formed by the line from the point (x+1, y-1) to the origin. This suggests a geometric relationship where these angles intersect.
Step 4
Evaluate the options
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Answer
By examining the diagrams provided, the correct diagram will illustrate the relationship between these angles. The solution must show a line that represents all points z such that the angles are equivalent.
Step 5
Select the correct diagram
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Answer
After analyzing the diagrams, the correct answer is D, which clearly represents the solution set formed by the angles being equal.
