The number of values of z ∈ ℂ , satisfying the equations...

The first equation $|z-(4+8i)|=\sqrt{10}$ represents a circle with center $C(4, 8)$ and radius $r = \sqrt{10}$.

The second equation $|z-(3+5i)|+|z-(5+11i)|=4\sqrt{5}$ represents an ellipse with foci at $S_1(3, 5) and S_2(5, 11)$, and length of the major axis $2a = 4\sqrt{5}$.

The distance between the foci is $2ae = \sqrt{(5-3)^2 + (11-5)^2} = \sqrt{4+36} = \sqrt{40} = 2\sqrt{10}$.

The center of the ellipse is the midpoint of the line segment joining the foci, which is $\left(\dfrac{3+5}{2}, \dfrac{5+11}{2}\right) = (4, 8)$. This is the same as the center of the circle.

The semi-major axis is $a = 2\sqrt{5}$, so $a^2 = 20$.

The semi-minor axis $b$ is given by $b^2 = a^2 - (ae)^2 = 20 - 10 = 10$, which gives $b = \sqrt{10}$.

Since the circle and the ellipse are concentric and the radius of the circle $r = \sqrt{10}$ is equal to the semi-minor axis $b$ of the ellipse, the circle is inscribed in the ellipse and touches it exactly at the two endpoints of the minor axis.

Therefore, there are exactly $2$ points of intersection, meaning there are $2$ values of $z$ satisfying both equations.

Answer: $2$