If ` 2 | z - 1| = | z - 2| ` and if ` x^(2) + y^(2) = lambda x , "then " lambda = `

Options

A. `1//3`
B. `2//3`
C. `4//3`
D. none

Answer

To solve the problem, we start with the given equation involving the complex number \( z \):

1. **Given Equation**:
\[ 2 | z - 1 | = | z - 2 | \]

2. **Let \( z = x + iy \)**:
Here, \( x \) is the real part and \( y \) is the imaginary part of the complex number \( z \).

3. **Rewrite the Modulus**:
\[ | z - 1 | = | (x + iy) - 1 | = | (x - 1) + iy | = \sqrt{(x - 1)^2 + y^2} \]
\[ | z - 2 | = | (x + iy) - 2 | = | (x - 2) + iy | = \sqrt{(x - 2)^2 + y^2} \]

4. **Substituting into the Given Equation**:
\[ 2 \sqrt{(x - 1)^2 + y^2} = \sqrt{(x - 2)^2 + y^2} \]

5. **Squaring Both Sides**:
\[ 4((x - 1)^2 + y^2) = (x - 2)^2 + y^2 \]

6. **Expanding Both Sides**:
- Left Side:
\[ 4((x - 1)^2 + y^2) = 4(x^2 - 2x + 1 + y^2) = 4x^2 - 8x + 4 + 4y^2 \]
- Right Side:
\[ (x - 2)^2 + y^2 = (x^2 - 4x + 4 + y^2) = x^2 - 4x + 4 + y^2 \]

7. **Setting Both Sides Equal**:
\[ 4x^2 - 8x + 4 + 4y^2 = x^2 - 4x + 4 + y^2 \]

8. **Rearranging the Equation**:
\[ 4x^2 - x^2 + 4y^2 - y^2 - 8x + 4x + 4 - 4 = 0 \]
\[ 3x^2 + 3y^2 - 4x = 0 \]

9. **Factoring Out Common Terms**:
\[ 3(x^2 + y^2) = 4x \]

10. **Dividing by 3**:
\[ x^2 + y^2 = \frac{4}{3}x \]

11. **Identifying \( \lambda \)**:
We know from the problem statement that:
\[ x^2 + y^2 = \lambda x \]
Therefore, we can equate:
\[ \lambda x = \frac{4}{3}x \]
Thus, we find:
\[ \lambda = \frac{4}{3} \]

12. **Final Answer**:
The value of \( \lambda \) is:
\[ \lambda = \frac{4}{3} \]