Two points P and Q in the argand plane represent the complex numbers z and `3z+2+u`. If `|z|=2`, then Q moves on the circle, whose centre and radius are (here, `i^(2)=-1`)

Options

A. `-2+i,6`
B. `2-i,3`
C. `2+i,6`
D. `2+i,2`

Answer

To solve the problem, we need to find the center and radius of the circle on which the point Q moves in the Argand plane, given the complex numbers \( z \) and \( 3z + 2 + i \).

### Step-by-step Solution:

1. **Understanding the Given Information**:
We have two complex numbers:
- \( z \) (represented by point P)
- \( 3z + 2 + i \) (represented by point Q)
We also know that \( |z| = 2 \).

2. **Expressing \( z \)**:
Since \( |z| = 2 \), we can express \( z \) in terms of its real and imaginary parts:
\[ z = x + iy \]
where \( x^2 + y^2 = 4 \) (because \( |z|^2 = x^2 + y^2 = 2^2 = 4 \)).

3. **Finding Q**:
Now, substituting \( z \) into the expression for Q:
\[ Q = 3z + 2 + i = 3(x + iy) + 2 + i = 3x + 2 + (3y + 1)i \]
Thus, the coordinates of Q in the Argand plane are:
\[ Q = (3x + 2) + (3y + 1)i \]

4. **Setting Up the Circle Equation**:
Let \( h = 3x + 2 \) and \( k = 3y + 1 \). We can express \( x \) and \( y \) in terms of \( h \) and \( k \):
\[ x = \frac{h - 2}{3}, \quad y = \frac{k - 1}{3} \]

5. **Substituting into the Circle Equation**:
Since \( |z|^2 = 4 \), we substitute \( x \) and \( y \) into the equation:
\[ \left(\frac{h - 2}{3}\right)^2 + \left(\frac{k - 1}{3}\right)^2 = 4 \]
Multiplying through by 9 gives:
\[ (h - 2)^2 + (k - 1)^2 = 36 \]

6. **Expanding the Equation**:
Expanding the equation:
\[ (h^2 - 4h + 4) + (k^2 - 2k + 1) = 36 \]
This simplifies to:
\[ h^2 + k^2 - 4h - 2k + 5 = 36 \]
Rearranging gives:
\[ h^2 + k^2 - 4h - 2k - 31 = 0 \]

7. **Identifying the Center and Radius**:
This is the equation of a circle in the form:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
where:
- \( g = -2 \)
- \( f = -1 \)
- \( c = -31 \)

The center \( (h, k) \) is given by:
\[ \left(-g, -f\right) = (2, 1) \]
The radius \( r \) is given by:
\[ r = \sqrt{g^2 + f^2 - c} = \sqrt{(-2)^2 + (-1)^2 + 31} = \sqrt{4 + 1 + 31} = \sqrt{36} = 6 \]

### Final Answer:
- The center of the circle is \( (2, 1) \) or \( 2 + i \).
- The radius of the circle is \( 6 \).