Which diagram best represents the solutions to the equation $ ext{arg}(z) = ext{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 $ ext{arg}(z) = ext{arg}(z + 1 - i)$?
A.
B.
C.
D.
Worked Solution & Example Answer:Which diagram best represents the solutions to the equation $ ext{arg}(z) = ext{arg}(z + 1 - i)$?
A - HSC - SSCE Mathematics Extension 2 - Question 7 - 2018 - Paper 1
Step 1
Determine the condition for equality of arguments
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Answer
The equation extarg(z)=extarg(z+1−i) implies that the angle (or argument) formed by the complex number z must equal the angle formed by z+1−i. This means both points must lie on the same line when represented on the Argand plane (complex plane).
Step 2
Analyze the geometric representation
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Answer
To find the appropriate diagram, we can represent z as a point (x,y) in the complex plane. The equation can be interpreted to mean that the line representing z has to intersect with the line created by the point (−1,1), derived from the expression z+1−i. Thus, the arguments being equal signifies that the angle from the origin to point z is equal to the angle from the origin to the point (−1,1).
Step 3
Identify the correct diagram
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Answer
Among the options presented, Diagram D shows the correct representation of the relationship described. It illustrates a line that goes through the point (−1,1) and through the origin, indicating that the arguments of both complex numbers are equal. Hence, the correct choice is D.
