Question:

Step 1: Interpret the geometric meaning of the equations. The equation $|z| = 3$ represents a circle $C_1$ centered at the origin $O(0,0)$ with radius $r_1 = 3$. The equation $|z + 1 - i| = 2$ can be rewritten as $|z - (-1 + i)| = 2$, which represents a circle $C_2$ centered at the point $P(-1, 1)$ with radius $r_2 = 2$. Step 2: Calculate the distance between the two centers. The distance $d$ between the centers $O(0,0)$ and $P(-1,1)$ is given by: $$d = |(-1 + i) - (0 + 0i)| = |-1 + i| = \sqrt{(-1)^2 + (1)^2} = \sqrt{1 + 1} = \sqrt{2}$$ Step 3: Apply the condition for the intersection of two circles. Two circles intersect if and only if $|r_1 - r_2| \le d \le r_1 + r_2$. Substituting the values: $$|3 - 2| \le \sqrt{2} \le 3 + 2$$ $$1 \le \sqrt{2} \le 5$$ Since $\sqrt{2} \approx 1.414$, the inequality $1 \le 1.414 \le 5$ is true. This condition indicates that the circles intersect. Step 4: Determine the nature of intersection. The conditions for the number of solutions are: * If $d = r_1 + r_2$: one solution (externally tangent). * If $|r_1 - r_2| < d < r_1 + r_2$: two distinct solutions. * If $d = |r_1 - r_2|$: one solution (internally tangent). * If $d < |r_1 - r_2|$ or $d > r_1 + r_2$: no solution. In this case, $d = \sqrt{2} \approx 1.414$, $|r_1 - r_2| = 1$, and $r_1 + r_2 = 5$. Since $1 < \sqrt{2} < 5$, the condition $|r_1 - r_2| < d < r_1 + r_2$ is satisfied. This implies that the two circles intersect at exactly two distinct points. Therefore, the system of equations has two solutions.