Question

The number of points of intersection of $|z - (4 + 3i)| = 2$ and $|z| + |z - 4| = 6$, $z \in \mathbb{C}$, is 

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

The Correct Option is C

To solve this problem, we need to find the number of intersection points between the given equations:

1. Equation of a circle in the complex plane: |z - (4 + 3i)| = 2
2. Equation of the ellipse (or line segment boundaries in this context): |z| + |z - 4| = 6

Let's analyze each component:

1. The equation |z - (4 + 3i)| = 2 represents a circle with center at the point (4, 3) and radius 2.

2. The equation |z| + |z - 4| = 6 represents the set of points in the complex plane such that the sum of distances from the origin and the point (4, 0) is 6. This describes an ellipse with foci at (0, 0) and (4, 0), where the sum of distances from any point on the ellipse to these two foci is constant and equal to 6.

Now, let's determine the points of intersection:

1. First, find the points on the circle, centered at (4, 3) with radius 2. Any point (x, y) on this circle satisfies:

   (x - 4)^2 + (y - 3)^2 = 4

2. For the ellipse condition, we know:

   |z| + |z - 4| = 6

   Let z = x + yi. The equation becomes:

   \sqrt{x^2 + y^2} + \sqrt{(x - 4)^2 + y^2} = 6

3. Substitute the circle's equation into the ellipse equation to determine possible solutions for (x, y).

   By solving these equations simultaneously, we determine the possible intersection points.

Upon solving these equations, the number of distinct points of intersection between the circle and the ellipse is 2.

Thus, the correct answer is 2.