Question

All the points in $A=\{\frac{\lambda+i}{\lambda-i}; \lambda \in \mathbb{R}\}$ lie on ________.

• A. a circle with radius $\sqrt{2}$
• B. a circle with radius 2
• C. a circle with radius $\frac{1}{2}$
• D. a circle with radius 1
• E. a straight line with slope 1

Hint:

The modulus of $\frac{a+bi}{a-bi}$ is always 1.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
We are given a set of complex numbers, and we need to find the geometric locus of these points in the complex plane. A common technique is to find the modulus of the complex number. If the modulus is a constant value 'r', the locus is a circle of radius 'r' centered at the origin.
Step 2: Key Formula or Approach
Let \(z\) be a complex number from the set A.
\[ z = \frac{\lambda + i}{\lambda - i} \] We will find the modulus of \(z\), denoted as \(|z|\). We will use the properties of moduli:
1. \(|z_1 / z_2| = |z_1| / |z_2|\)
2. For a complex number \(a + bi\), its modulus is \(|a + bi| = \sqrt{a^2 + b^2}\).
Step 3: Detailed Explanation
1. Set up the modulus calculation.
\[ |z| = \left| \frac{\lambda + i}{\lambda - i} \right| \] 2. Apply the property of modulus of a quotient.
\[ |z| = \frac{|\lambda + i|}{|\lambda - i|} \] 3. Calculate the modulus of the numerator and the denominator.
Since \(\lambda\) is a real number, we can treat \(\lambda + i\) as \(\lambda + 1i\) and \(\lambda - i\) as \(\lambda - 1i\).
Modulus of the numerator:
\[ |\lambda + i| = \sqrt{(\lambda)^2 + (1)^2} = \sqrt{\lambda^2 + 1} \] Modulus of the denominator:
\[ |\lambda - i| = \sqrt{(\lambda)^2 + (-1)^2} = \sqrt{\lambda^2 + 1} \] 4. Compute the final modulus of z.
\[ |z| = \frac{\sqrt{\lambda^2 + 1}}{\sqrt{\lambda^2 + 1}} = 1 \] 5. Interpret the result.
The result \(|z| = 1\) means that for any real value of \(\lambda\), the complex number \(z\) will always have a modulus of 1. In the complex plane, the equation \(|z| = r\) represents a circle centered at the origin with radius \(r\).
Therefore, all points \(z\) lie on a circle centered at the origin with a radius of 1.
Step 4: Final Answer
The points lie on a circle with radius 1.