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(C) The equation $|z - (4 + 3i)| = 2$ represents a circle with center $(4, 3)$ and radius $r = 2$.In Cartesian coordinates,this is $(x - 4)^2 + (y - 3

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$2$

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(C) The equation $|z - (4 + 3i)| = 2$ represents a circle with center $(4, 3)$ and radius $r = 2$.In Cartesian coordinates,this is $(x - 4)^2 + (y - 3)^2 = 4$.The equation $|z| + |z - 4| = 6$ represents an ellipse with foci at $(0, 0)$ and $(4, 0)$.The sum of distances from the foci is $2a = 6$,so $a = 3$. The center is $(2, 0)$.The distance between foci is $2ae = 4$,so $ae = 2$. Since $a = 3$,$e = 2/3$.$b^2 = a^2(1 - e^2) = 9(1 - 4/9) = 5$.The equation of the ellipse is $\frac{(x - 2)^2}{9} + \frac{y^2}{5} = 1$.Check the position of the circle relative to the ellipse.The center of the circle is $(4, 3)$. The highest point of the ellipse is $(2, \sqrt{5}) \approx (2, 2.236)$.The circle's lowest point is $(4, 3 - 2) = (4, 1)$.Substitute $(4, 1)$ into the ellipse equation: $\frac{(4 - 2)^2}{9} + \frac{1^2}{5} = \frac{4}{9} + \frac{1}{5} = \frac{20 + 9}{45} = \frac{29}{45} Since the point $(4, 1)$ lies inside the ellipse and the center $(4, 3)$ lies outside the ellipse,the circle must intersect the ellipse at two points.

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

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