Number of Complex Numbers Satisfying Modulus Equations

To find the number of complex numbers $z \in \mathbb{C}$ satisfying the given equations, let us analyze them geometrically in the complex plane where $z = x + iy$.

1. Analysis of the First Equation: $|z - (4 + 8i)| = \sqrt{10}$ This equation represents a circle $C$ in the complex plane with:

• Center: $C_0 = (4, 8)$
• Radius: $R = \sqrt{10}$

2. Analysis of the Second Equation: $|z - (3 + 5i)| + |z - (5 + 11i)| = 4\sqrt{5}$ This equation represents an ellipse $E$ with foci at $A(3, 5)$ and $B(5, 11)$.

• Center of the ellipse ($M$): The center of the ellipse is the midpoint of $A$ and $B$: $M = \left(\frac{3 + 5}{2}, \frac{5 + 11}{2}\right) = (4, 8)$ Notice that the center of the ellipse coincides with the center of the circle $C_0(4, 8)$.

• Distance between foci ($2ae$): $2ae = AB = \sqrt{(5 - 3)^2 + (11 - 5)^2} = \sqrt{2^2 + 6^2} = \sqrt{40} = 2\sqrt{10} \implies ae = \sqrt{10}$

• Length of the major axis ($2a$): $2a = 4\sqrt{5} \implies a = 2\sqrt{5}$ Thus, $a^2 = 20$.

• Length of the semi-minor axis ($b$): $b = \sqrt{a^2 - (ae)^2} = \sqrt{20 - 10} = \sqrt{10}$

3. Points of Intersection: In a local coordinate system $(x', y')$ centered at $(4, 8)$ and rotated along the major axis of the ellipse:

• The equation of the circle is: $x'^2 + y'^2 = R^2 = 10$
• The equation of the ellipse is: $\frac{x'^2}{a^2} + \frac{y'^2}{b^2} = 1 \implies \frac{x'^2}{20} + \frac{y'^2}{10} = 1$

Substituting $y'^2 = 10 - x'^2$ from the circle into the ellipse equation gives: $\frac{x'^2}{20} + \frac{10 - x'^2}{10} = 1 \implies \frac{x'^2}{20} + 1 - \frac{x'^2}{10} = 1 \implies -\frac{x'^2}{20} = 0 \implies x' = 0$

For $x' = 0$, we get $y' = \pm \sqrt{10}$.

Since $R = b = \sqrt{10}$, the circle touches the ellipse internally at the two endpoints of its minor axis, $(0, \sqrt{10})$ and $(0, -\sqrt{10})$.

Thus, there are exactly $2$ points of intersection.

Correct Answer: B ($2$)