The number of values of z in mathbb{C}, satis | vedclass

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(B) The first equation $|z - (4 + 8i)| = \sqrt{10}$ represents a circle with center $C(4, 8)$ and radius $r = \sqrt{10}$.The second equation $|z - (3

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$2$

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(B) The first equation $|z - (4 + 8i)| = \sqrt{10}$ represents a circle with center $C(4, 8)$ and radius $r = \sqrt{10}$.The second equation $|z - (3 + 5i)| + |z - (5 + 11i)| = 4\sqrt{5}$ represents an ellipse with foci $F_1(3, 5)$ and $F_2(5, 11)$.The distance between the foci is $2ae = \sqrt{(5-3)^2 + (11-5)^2} = \sqrt{2^2 + 6^2} = \sqrt{40} = 2\sqrt{10}$.The length of the major axis is $2a = 4\sqrt{5}$, so $a = 2\sqrt{5}$.The eccentricity $e$ is given by $2ae = 2\sqrt{10} \implies e = \frac{2\sqrt{10}}{2(2\sqrt{5})} = \frac{\sqrt{2}}{2} = \frac{1}{\sqrt{2}}$.The center of the ellipse is the midpoint of $F_1F_2$, which is $(\frac{3+5}{2}, \frac{5+11}{2}) = (4, 8)$. This matches the center of the circle.The semi-minor axis $b$ is given by $b^2 = a^2(1 - e^2) = (2\sqrt{5})^2(1 - 1/2) = 20(1/2) = 10$, so $b = \sqrt{10}$.Since the radius of the circle $r = \sqrt{10}$ is equal to the semi-minor axis $b = \sqrt{10}$, the circle is tangent to the ellipse at the two endpoints of the minor axis.Therefore, there are $2$ values of $z$ that satisfy both equations.

The number of values of $z \in \mathbb{C}$, satisfying the equations $|z - (4 + 8i)| = \sqrt{10}$ and $|z - (3 + 5i)| + |z - (5 + 11i)| = 4\sqrt{5}$, is:

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