Problem Analysis #14

[THOUGHT] To find the parameters of the circle ( S ), I need to first translate the given equation ( |x + iy - z_1| = 2|x + iy - z_2| ) into a standard circle equation.

Given: [ z_1 = 1 + 2i ] [ z_2 = 3i ] [ S = {(x, y) \in \mathbb{R} \times \mathbb{R} : |x + iy - z_1| = 2|x + iy - z_2|} ]

Let ( z = x + iy ). Then the equation becomes: [ |z - z_1| = 2|z - z_2| ]

Substitute ( z_1 ) and ( z_2 ): [ |x + iy - (1 + 2i)| = 2|x + iy - 3i| ] [ |(x - 1) + i(y - 2)| = 2|(x) + i(y - 3)| ]

The modulus of a complex number ( a + bi ) is ( \sqrt{a^2 + b^2} ). Therefore, we have: [ \sqrt{(x - 1)^2 + (y - 2)^2} = 2\sqrt{x^2 + (y - 3)^2} ]

Square both sides to eliminate the square roots: [ (x - 1)^2 + (y - 2)^2 = 4(x^2 + (y - 3)^2) ]

Expand both sides: [ (x^2 - 2x + 1) + (y^2 - 4y + 4) = 4(x^2 + y^2 - 6y + 9) ] [ x^2 - 2x + 1 + y^2 - 4y + 4 = 4x^2 + 4y^2 - 24y + 36 ]

Combine like terms: [ x^2 + y^2 - 2x - 4y + 5 = 4x^2 + 4y^2 - 24y + 36 ]

Move all terms to one side: [ x^2 + y^2 - 2x - 4y + 5 - 4x^2 - 4y^2 + 24y - 36 = 0 ] [ -3x^2 - 3y^2 - 2x + 20y - 31 = 0 ]

Divide the entire equation by (-3) to simplify: [ x^2 + y^2 + \frac{2}{3}x - \frac{20}{3}y + \frac{31}{3} = 0 ]

To convert this into the standard circle equation form ((x - x_0)^2 + (y - y_0)^2 = r^2), we need to complete the square for (x) and (y).

For (x): [ x^2 + \frac{2}{3}x = \left(x + \frac{1}{3}\right)^2 - \left(\frac{1}{3}\right)^2 = \left(x + \frac{1}{3}\right)^2 - \frac{1}{9} ]

For (y): [ y^2 - \frac{20}{3}y = \left(y - \frac{10}{3}\right)^2 - \left(\frac{10}{3}\right)^2 = \left(y - \frac{10}{3}\right)^2 - \frac{100}{9} ]

Substitute back into the equation: [ \left(x + \frac{1}{3}\right)^2 - \frac{1}{9} + \left(y - \frac{10}{3}\right)^2 - \frac{100}{9} + \frac{31}{3} = 0 ] [ \left(x + \frac{1}{3}\right)^2 + \left(y - \frac{10}{3}\right)^2 - \frac{101}{9} + \frac{93}{9} = 0 ] [ \left(x + \frac{1}{3}\right)^2 + \left(y - \frac{10}{3}\right)^2 - \frac{8}{9} = 0 ] [ \left(x + \frac{1}{3}\right)^2 + \left(y - \frac{10}{3}\right)^2 = \frac{8}{9} ]

From this equation, we identify the center ((x_0, y_0)) and the radius (r): [ x_0 = -\frac{1}{3}, \quad y_0 = \frac{10}{3}, \quad r^2 = \frac{8}{9} ]

Thus, ( r = \sqrt{\frac{8}{9}} = \frac{2\sqrt{2}}{3} ).

Now, calculate ( 9(x_0 + y_0)r^2 ): [ 9\left(-\frac{1}{3} + \frac{10}{3}\right)\left(\frac{8}{9}\right) ] [ 9\left(\frac{9}{3}\right)\left(\frac{8}{9}\right) ] [ 9 \cdot 3 \cdot \frac{8}{9} ] [ 3 \cdot 8 = 24 ]

Therefore, the value of ( 9(x_0 + y_0)r^2 ) is: [ \boxed{24} ] [/THOUGHT]