Key Concepts and Formulas

• Geometric Interpretation of Complex Numbers: The modulus $|z - z_0|$ represents the distance between the complex number $z$ and the complex number $z_0$ in the complex plane.
• Equation of a Circle: An equation of the form $|z - z_0| = R$ represents a circle with center $z_0$ and radius $R$.
• Equation of an Ellipse: An equation of the form $|z - F_1| + |z - F_2| = 2a$ represents an ellipse with foci at $F_1$ and $F_2$, and major axis length $2a$. The relationship between the semi-major axis $a$, semi-minor axis $b$, and distance from the center to each focus $c$ is given by $a^2 = b^2 + c^2$.

Step-by-Step Solution

Step 1: Analyze the first equation: The Circle

The first equation is: $|z - (4 + 3i)| = 2$

This equation represents a circle in the complex plane. We are identifying the geometric shape represented by the given complex equation. The modulus signifies distance, and the equation states that the distance between a complex number $z$ and the complex number $4+3i$ is constant and equal to 2.

• Center of the circle: $C = (4, 3)$
• Radius of the circle: $R = 2$

Step 2: Analyze the second equation: The Ellipse

The second equation is: $|z| + |z - 4| = 6$

This equation represents an ellipse in the complex plane. We are identifying the geometric shape. The equation states that the sum of the distances from any point $z$ to two fixed points is constant. This is the definition of an ellipse.

• Foci of the ellipse: $F_1 = (0, 0)$ and $F_2 = (4, 0)$
• Major axis length: $2a = 6$, so $a = 3$
• Distance between foci: $2c = |4 - 0| = 4$, so $c = 2$
• Semi-minor axis: Using $a^2 = b^2 + c^2$, we have $3^2 = b^2 + 2^2$, which gives $9 = b^2 + 4$, so $b^2 = 5$ and $b = \sqrt{5}$
• Center of the ellipse: The midpoint of the foci is $E = (\frac{0+4}{2}, \frac{0+0}{2}) = (2, 0)$

Step 3: Determine the position of the circle's center relative to the ellipse

To understand the intersection possibilities, we need to know if the circle's center lies inside or outside the ellipse. We can check this by calculating the sum of the distances from the circle's center to the ellipse's foci. If this sum is less than $2a = 6$, the center is inside; if it's greater, the center is outside.

• Distance from $C(4, 3)$ to $F_1(0, 0)$: $d_1 = \sqrt{(4-0)^2 + (3-0)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$
• Distance from $C(4, 3)$ to $F_2(4, 0)$: $d_2 = \sqrt{(4-4)^2 + (3-0)^2} = \sqrt{0 + 9} = \sqrt{9} = 3$
• Sum of distances: $d_1 + d_2 = 5 + 3 = 8$

Since $8 > 6$, the center of the circle $(4, 3)$ lies outside the ellipse.

Step 4: Determine if any points on the ellipse lie inside the circle

To further refine our understanding of the intersection, we need to determine if any points on the ellipse lie inside the circle. Let's consider points on the ellipse where $x=4$. We substitute $x=4$ into the ellipse equation $\frac{(x-2)^2}{a^2} + \frac{y^2}{b^2} = 1$: $\frac{(4-2)^2}{3^2} + \frac{y^2}{(\sqrt{5})^2} = 1$ $\frac{4}{9} + \frac{y^2}{5} = 1$ $\frac{y^2}{5} = 1 - \frac{4}{9} = \frac{5}{9}$ $y^2 = \frac{25}{9} \implies y = \pm \frac{5}{3}$

So, the ellipse passes through the points $(4, 5/3)$ and $(4, -5/3)$.

Now, let's check these two points against the circle with center $(4,3)$ and radius $2$:

• Distance from $(4, 5/3)$ to $C(4,3) = \sqrt{(4-4)^2 + (5/3 - 3)^2} = \sqrt{0 + (5/3 - 9/3)^2} = \sqrt{(-4/3)^2} = \sqrt{16/9} = 4/3$. Since $4/3 < 2$, the point $(4, 5/3)$ lies inside the circle.
• Distance from $(4, -5/3)$ to $C(4,3) = \sqrt{(4-4)^2 + (-5/3 - 3)^2} = \sqrt{0 + (-5/3 - 9/3)^2} = \sqrt{(-14/3)^2} = \sqrt{196/9} = 14/3$. Since $14/3 > 2$, the point $(4, -5/3)$ lies outside the circle.

Step 5: Analyze the number of intersections Since the center of the circle is outside the ellipse, but a point on the ellipse $(4, 5/3)$ lies inside the circle, there must be at least two points of intersection. The ellipse is a closed, continuous curve. As we move along the ellipse from $(4, -5/3)$ (outside the circle) to $(4, 5/3)$ (inside the circle), we must cross the circle's boundary. Also, the distance from $(5,0)$ to $(4,3)$ is $\sqrt{10}>2$ so $(5,0)$ is outside the circle. As we continue from $(4, 5/3)$ (inside) to $(5,0)$ (outside), we must again cross the circle's boundary.

Let's consider the range of $y$ values. The circle is centered at $y=3$ with radius 2, therefore the circle's $y$ values range from $1$ to $5$. The ellipse has semi-minor axis $\sqrt{5} \approx 2.236$. The upper co-vertex is at $y=\sqrt{5}$, and the lower co-vertex is at $y=-\sqrt{5}$. Thus, only the upper portion of the ellipse intersects the circle.

Because the point $(4, 5/3)$ is inside the circle, and the point $(5,0)$ is outside the circle, and the center of the circle is $(4,3)$ and radius is 2, and the circle lies entirely above $y=1$, the two intersection points must lie above the x-axis. The lower half of the ellipse is completely outside the circle.

Therefore, there are two points of intersection.

Common Mistakes & Tips

• Incorrectly determining whether the center of the circle lies inside or outside the ellipse.
• Assuming no intersection without carefully analyzing the relative positions of the curves and testing specific points.
• Algebra errors in calculating distances or solving equations.

Summary

By analyzing the equations as a circle and an ellipse and determining the relative positions of their centers and key points, we found that the ellipse intersects the circle at two distinct points. The upper half of the ellipse intersects the circle twice, while the lower half does not intersect at all.

Final Answer

The final answer is \boxed{2}, which corresponds to option (C).