Question:
If $|z + 4| = 2|z + 1|$, where $z$ is a complex number then $|z|$ is equal to
If $|z + 4| = 2|z + 1|$, where $z$ is a complex number then $|z|$ is equal to
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Convert modulus equations into coordinate form and simplify.
Updated On: Apr 30, 2026
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The Correct Option is B
Solution and Explanation
Concept:
Use geometric interpretation of complex numbers.
Step 1: Let $z = x + iy$
\[ |z + 4| = \sqrt{(x+4)^2 + y^2}, \quad |z + 1| = \sqrt{(x+1)^2 + y^2} \]
Step 2: Square both sides
\[ (x+4)^2 + y^2 = 4[(x+1)^2 + y^2] \]
Step 3: Expand
\[ x^2 + 8x + 16 + y^2 = 4(x^2 + 2x + 1 + y^2) \] \[ x^2 + 8x + 16 + y^2 = 4x^2 + 8x + 4 + 4y^2 \]
Step 4: Simplify
\[ 0 = 3x^2 + 3y^2 - 12 \] \[ x^2 + y^2 = 4 \]
Step 5: Interpret
\[ |z| = \sqrt{x^2 + y^2} = 2 \] Final Conclusion:
Option (B)
Step 1: Let $z = x + iy$
\[ |z + 4| = \sqrt{(x+4)^2 + y^2}, \quad |z + 1| = \sqrt{(x+1)^2 + y^2} \]
Step 2: Square both sides
\[ (x+4)^2 + y^2 = 4[(x+1)^2 + y^2] \]
Step 3: Expand
\[ x^2 + 8x + 16 + y^2 = 4(x^2 + 2x + 1 + y^2) \] \[ x^2 + 8x + 16 + y^2 = 4x^2 + 8x + 4 + 4y^2 \]
Step 4: Simplify
\[ 0 = 3x^2 + 3y^2 - 12 \] \[ x^2 + y^2 = 4 \]
Step 5: Interpret
\[ |z| = \sqrt{x^2 + y^2} = 2 \] Final Conclusion:
Option (B)
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