Question:
If the amplitude of \(z - 2 - 3i\) is \(\pi/4\), then the locus of \(z = x + i y\) is:
If the amplitude of \(z - 2 - 3i\) is \(\pi/4\), then the locus of \(z = x + i y\) is:
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For argument π/4, imaginary and real parts are equal.
Updated On: Mar 23, 2026
- \(x+y-1=0\)
- \(x-y-1=0\)
- \(x+y+1=0\)
- x-y+1=0
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The Correct Option is A
Solution and Explanation
Step 1: Let \(z = x + i y\). Then
\[ z - 2 - 3i = (x - 2) + i(y - 3) \]
Step 2: Given \(\arg(z - 2 - 3i) = \pi/4\), so
\[ \frac{y - 3}{x - 2} = 1 \]
Step 3: Thus
\[ y - 3 = x - 2 \implies x + y - 1 = 0 \]
\[ z - 2 - 3i = (x - 2) + i(y - 3) \]
Step 2: Given \(\arg(z - 2 - 3i) = \pi/4\), so
\[ \frac{y - 3}{x - 2} = 1 \]
Step 3: Thus
\[ y - 3 = x - 2 \implies x + y - 1 = 0 \]
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