The number of values of $$z \in \mathbb{C}$$, satisfying the equations $$|z - (4 + 8i)| = \sqrt{10}$$ and $$|z - (3 + 5i)| + |z - (5 + 11i)| = 4\sqrt{5}$$, is :

This is a great geometry-based complex numbers problem. To find the number of values of $$z$$, we need to identify the geometric shapes represented by these two equations and see how many times they intersect.

Step 1: Identify the First Shape

The equation $$|z - (4 + 8i)| = \sqrt{10}$$ represents a circle.

• Center ($$C$$): $$(4, 8)$$
• Radius ($$r$$): $$\sqrt{10} \approx 3.16$$
• Foci: $$F_1(3, 5)$$ and $$F_2(5, 11)$$
• Distance between foci ($$2ae$$):
• Length of major axis ($$2a$$): $$4\sqrt{5} = \sqrt{80}$$
• Midpoint = $$(\frac{3+5}{2}, \frac{5+11}{2}) = (4, 8)$$
• Semi-major axis ($$a$$): $$\frac{4\sqrt{5}}{2} = 2\sqrt{5} = \sqrt{20} \approx 4.47$$
• Distance from center to focus ($$ae$$): $$\frac{2\sqrt{10}}{2} = \sqrt{10} \approx 3.16$$
• Semi-minor axis ($$b$$): Using $$b^2 = a^2 - (ae)^2$$:

Step 2: Identify the Second Shape

The equation $$|z - (3 + 5i)| + |z - (5 + 11i)| = 4\sqrt{5}$$ represents an ellipse, provided that the constant ($$4\sqrt{5}$$) is greater than the distance between the two fixed points (foci).

$$\sqrt{(5-3)^2 + (11-5)^2} = \sqrt{2^2 + 6^2} = \sqrt{4 + 36} = \sqrt{40} = 2\sqrt{10}$$

Since $$\sqrt{80} > \sqrt{40}$$ (i.e., $$2a > 2ae$$), this is a valid ellipse.

Step 3: Check the Relationship

Interestingly, let's find the midpoint of the foci of the ellipse:

Notice: The center of the circle is exactly the same as the center of the ellipse! Both are centered at $$(4, 8)$$.

Step 4: Determine Intersection

To see if they intersect, let's compare the radius of the circle to the semi-minor and semi-major axes of the ellipse.

$$b^2 = 20 - 10 = 10 \implies b = \sqrt{10}$$

Conclusion:

The radius of the circle ($$r = \sqrt{10}$$) is exactly equal to the semi-minor axis of the ellipse ($$b = \sqrt{10}$$).

Because the circle is centered at the same point as the ellipse and its radius falls between the length of the semi-minor and semi-major axes (specifically, it equals the semi-minor axis), the circle will be tangent to the ellipse at the two co-vertices.

Therefore, there are 2 points of intersection.

Correct Option: B (2)