The user wants me to solve a problem involving the locus of a complex number, which is a common topic in JEE mathematics.

Step-by-Step Solution:

1. Understand the Given Equation: We are given the set S of points (x, y) in the real plane ℝ × ℝ. Each point (x, y) corresponds to a complex number z = x + iy. The condition for a point to be in S is: $|x + iy - z_1| = 2|x + iy - z_2|$ where $z_1 = 1 + 2i$ and $z_2 = 3i$. In terms of the complex number z, this equation is: $|z - z_1| = 2|z - z_2|$ This equation describes the locus of points z such that the distance from z to $z_1$ is twice the distance from z to $z_2$. This is the definition of a Circle of Apollonius.

2. Substitute the Complex Numbers and Simplify: Let z = x + iy. Substitute the given values of $z_1$ and $z_2$ into the equation: $| (x + iy) - (1 + 2i) | = 2 | (x + iy) - 3i |$ Group the real and imaginary parts inside the modulus: $| (x - 1) + i(y - 2) | = 2 | x + i(y - 3) |$

3. Use the Definition of Modulus: The modulus of a complex number a + ib is √(a² + b²). Applying this to our equation: $\sqrt{(x - 1)^2 + (y - 2)^2} = 2 \sqrt{x^2 + (y - 3)^2}$

4. Square Both Sides and Expand: To eliminate the square roots, we square both sides of the equation: $(x - 1)^2 + (y - 2)^2 = 4 [x^2 + (y - 3)^2]$ Now, expand the squared terms: $(x^2 - 2x + 1) + (y^2 - 4y + 4) = 4(x^2 + y^2 - 6y + 9)$ $x^2 + y^2 - 2x - 4y + 5 = 4x^2 + 4y^2 - 24y + 36$

5. Rearrange into the Standard Equation of a Circle: Move all terms to one side to get the general form of a circle's equation, Ax² + Ay² + Dx + Ey + F = 0. $0 = (4x^2 - x^2) + (4y^2 - y^2) + 2x + (4y - 24y) + (36 - 5)$ $0 = 3x^2 + 3y^2 + 2x - 20y + 31$ This is the equation of a circle. To find its center and radius, we can convert it to the standard form x² + y² + 2gx + 2fy + c = 0 by dividing the entire equation by 3: $x^2 + y^2 + \frac{2}{3}x - \frac{20}{3}y + \frac{31}{3} = 0$

6. Determine the Center and Radius: For a circle with the equation x² + y² + 2gx + 2fy + c = 0, the center is (-g, -f) and the radius is r = √(g² + f² - c).

   Comparing our equation with the standard form, we have: 2g = 2/3 => g = 1/3 2f = -20/3 => f = -10/3 c = 31/3

   The center of the circle is: (-g, -f) = (-1/3, -(-10/3)) = (-1/3, 10/3)

   The radius of the circle is: $r = \sqrt{(1/3)^2 + (-10/3)^2 - 31/3}$ $r = \sqrt{1/9 + 100/9 - 31/3}$ $r = \sqrt{101/9 - 93/9}$ $r = \sqrt{8/9}$ r = (√8) / (√9) = (2√2) / 3

7. Evaluate the Options:

   • A: S is a circle with centre (-1/3, 10/3) Our calculated center is (-1/3, 10/3). This statement is TRUE.
   • B: S is a circle with centre (1/3, 8/3) This does not match our calculated center. This statement is FALSE.
   • C: S is a circle with radius √2/3 This does not match our calculated radius. This statement is FALSE.
   • D: S is a circle with radius (2√2)/3 Our calculated radius is (2√2)/3. This statement is TRUE.

Conclusion:

Based on the calculations, the set S represents a circle with center (-1/3, 10/3) and radius (2√2)/3. Therefore, options A and D are correct.