The number of complex number z satisfying `abs(z-(4+3i))=2 and absz +abs(z-4) =6` is

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem, we need to analyze the two given equations involving complex numbers.

### Step 1: Understand the first equation
The first equation is given by:
\[ |z - (4 + 3i)| = 2 \]
This represents a circle in the complex plane. The center of the circle is at the point \( (4, 3) \) and the radius is \( 2 \).

### Step 2: Understand the second equation
The second equation is:
\[ |z| + |z - 4| = 6 \]
This represents an ellipse in the complex plane with foci at the points \( (0, 0) \) and \( (4, 0) \). The total distance from any point on the ellipse to the two foci is \( 6 \).

### Step 3: Identify the center and vertices of the ellipse
The foci of the ellipse are at \( (0, 0) \) and \( (4, 0) \). The distance between the foci is \( 4 \) (from \( 0 \) to \( 4 \)). The length of the major axis is \( 6 \), which means the distance from the center of the ellipse to its vertices is \( 3 \).

The center of the ellipse is at:
\[ \left( \frac{0 + 4}{2}, 0 \right) = (2, 0) \]
The vertices of the ellipse are at:
\[ (2 + 3, 0) = (5, 0) \quad \text{and} \quad (2 - 3, 0) = (-1, 0) \]

### Step 4: Draw the circle and the ellipse
Now, we can visualize the situation:
- The circle is centered at \( (4, 3) \) with a radius of \( 2 \).
- The ellipse is centered at \( (2, 0) \) with vertices at \( (5, 0) \) and \( (-1, 0) \).

### Step 5: Analyze the intersection points
Next, we need to find the intersection points of the circle and the ellipse.

1. The circle will intersect the line \( y = 0 \) (the x-axis) at two points, since the center of the circle is above the x-axis and the radius is sufficient to reach the x-axis.
2. The ellipse will also intersect the x-axis at two points, which are the vertices \( (5, 0) \) and \( (-1, 0) \).

### Step 6: Determine the number of intersection points
By sketching or calculating the distances, we can see that the circle intersects the ellipse at two distinct points.

Thus, the number of complex numbers \( z \) that satisfy both equations is:
\[ \boxed{2} \]