Let $$z = x + iy$$ where $$x, y \in \mathbb{R}$$.
$$\frac{z - 5}{z + 5} = \frac{(x - 5) + iy}{(x + 5) + iy} \cdot \frac{(x + 5) - iy}{(x + 5) - iy} = \frac{(x^2 + y^2 - 25) + i(10y)}{(x + 5)^2 + y^2}$$
Given $$\text{Re}\left(\frac{z - 5}{z + 5}\right) = 0$$:
$$\frac{x^2 + y^2 - 25}{(x + 5)^2 + y^2} = 0 \implies x^2 + y^2 = 25 \implies \vert z \vert = 5$$
This defines a circle centered at the origin with radius $$R = 5$$. Let the given fixed point be $$A = 3 + 4i$$. We check its position relative to the circle:
$$\vert A \vert = \sqrt{3^2 + 4^2} = 5$$
Since $$\vert A \vert = 5$$, the point $$A$$ lies directly on the circumference of the circle.
By circle geometry, the maximum distance between any two points on a circle occurs when they are diametrically opposite. Therefore, the point $$z_m$$ that maximizes the distance $$\vert z - A \vert$$ must be the opposite endpoint of the diameter passing through $$A$$.
$$z_m = -A = -(3 + 4i) = -3 - 4i$$
Extracting the real and imaginary parts of $$z_m$$:
$$\text{Re}(z_m) = -3, \quad \text{Im}(z_m) = -4$$
$$\text{Re}(z_m) \cdot \text{Im}(z_m) = (-3) \cdot (-4) = 12$$
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