Question:
The modulus of the complex number \( z \) such that \( |z + 3 - i| = 1 \) and \( \arg(z) = \pi \) is equal to:
The modulus of the complex number \( z \) such that \( |z + 3 - i| = 1 \) and \( \arg(z) = \pi \) is equal to:
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The modulus of a complex number \( z = a + bi \) is given by:
\[
|z| = \sqrt{a^2 + b^2}
\]
Updated On: Mar 26, 2025
- 3
- 2
- 9
- 4
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The Correct Option is C
Solution and Explanation
Step 1: {Convert given information into mathematical form}
\[ |z + 3 - i| = 1 \] Step 2: {Rewrite as a circle equation}
\[ (x + 3)^2 + (y - 1)^2 = 1 \] Step 3: {Find modulus}
Since \( \arg(z) = \pi \), the point lies on the negative real axis: \[ z = -3 + 0i \] Step 4: {Calculate modulus}
\[ |z| = \sqrt{(-3)^2 + 0^2} = 3 \]
\[ |z + 3 - i| = 1 \] Step 2: {Rewrite as a circle equation}
\[ (x + 3)^2 + (y - 1)^2 = 1 \] Step 3: {Find modulus}
Since \( \arg(z) = \pi \), the point lies on the negative real axis: \[ z = -3 + 0i \] Step 4: {Calculate modulus}
\[ |z| = \sqrt{(-3)^2 + 0^2} = 3 \]
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