If z is a complex number such that z + 1 + i = |z|, then which of the following statements are correct?
Understand the Problem
The question is asking about the properties of a complex number z given a specific equation involving z and the modulus of z. It requires analyzing the relationship between z and |z| to determine which statements are correct based on the condition provided.
Answer
A: True, D: True
Answer for screen readers
The correct statements are A and D:
$$ |z| = 1 \quad \text{and} \quad \arg(z) = \frac{\pi}{2} $$
Steps to Solve
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Rewrite the given equation
We start with the given equation:
$$ z + 1 + \frac{i}{|z|} = |z| $$ -
Isolate z terms
We rearrange the equation to isolate ( z ):
$$ z = |z| - 1 - \frac{i}{|z|} $$ -
Separate real and imaginary parts
Let ( z = x + iy ). Then the modulus ( |z| = \sqrt{x^2 + y^2} ).
Substituting ( z ) gives:
$$ x + iy = |z| - 1 - \frac{i}{|z|} $$
This gives us two separate equations:
Real part:
$$ x = |z| - 1 $$
Imaginary part:
$$ y = -\frac{1}{|z|} $$
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Substitute |z| in terms of x and y
Using ( |z| = \sqrt{x^2 + y^2} ), substitute ( |z| ) in the equations above.
From the real part equation:
$$ |z| = x + 1 $$
Substituting this into the modulus:
$$ x + 1 = \sqrt{x^2 + y^2} $$ -
Square both sides
Squaring both sides gives:
$$ (x + 1)^2 = x^2 + y^2 $$
Expanding:
$$ x^2 + 2x + 1 = x^2 + y^2 $$
This simplifies to:
$$ 2x + 1 = y^2 $$ -
Analyze the relationship
Now we can analyze the relationships among real and imaginary parts, satisfying the conditions for A, B, C, and D. -
Evaluate each option
- A: Check if ( |z| = 1 ) is possible.
- B: Check if ( z = \overline{z} ) (meaning ( y = 0 ) – real).
- C: Check if ( z = -\overline{z} ) (meaning ( x = 0 ) – purely imaginary).
- D: Calculate ( \arg(z) ).
The correct statements are A and D:
$$ |z| = 1 \quad \text{and} \quad \arg(z) = \frac{\pi}{2} $$
More Information
The modulus of a complex number and its argument are fundamental properties that help describe its position in the complex plane. The relationships derived show how various forms of ( z ) relate to each other geometrically.
Tips
- Failing to accurately isolate and equate the real and imaginary parts of ( z ).
- Not squaring both sides correctly or neglecting the modulus properties.
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