If |Z_1 - 3 - 4i| = 5 and |Z_2| = 15,then the | vedclass

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(B) Let $Z_0 = 3 + 4i$. The given equation $|Z_1 - Z_0| = 5$ represents a circle with center $C(3, 4)$ and radius $r = 5$.The distance of the center $

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$30$

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(B) Let $Z_0 = 3 + 4i$. The given equation $|Z_1 - Z_0| = 5$ represents a circle with center $C(3, 4)$ and radius $r = 5$.The distance of the center $C$ from the origin $O(0, 0)$ is $|Z_0| = \sqrt{3^2 + 4^2} = 5$.Since the radius is $5$ and the distance from the origin is $5$,the circle passes through the origin.Let $Z_2$ be a point such that $|Z_2| = 15$. The distance between the origin $O$ and $Z_2$ is $15$.By the triangle inequality,$|Z_1 - Z_2| \leq |Z_1| + |Z_2|$.The maximum value of $|Z_1|$ is the diameter of the circle,which is $2r = 10$ (since it passes through the origin,the maximum distance from the origin is the diameter).Thus,the maximum value of $|Z_1 - Z_2|$ is $10 + 15 = 25$.The minimum value of $|Z_1|$ is $0$ (since the circle passes through the origin).Thus,the minimum value of $|Z_1 - Z_2|$ is $|0 - 15| = 15$ is incorrect; rather,we consider the distance between the set of points on the circle and the point $Z_2$.The distance from the center $C$ to $Z_2$ ranges from $|OC - |Z_2||$ to $|OC + |Z_2||$,i.e.,$|5 - 15| = 10$ to $5 + 15 = 20$.Accounting for the radius $r=5$,the distance $|Z_1 - Z_2|$ ranges from $|10 - 5| = 5$ to $20 + 5 = 25$.The sum of the maximum and minimum values is $25 + 5 = 30$.

If $|Z_1 - 3 - 4i| = 5$ and $|Z_2| = 15$,then the sum of the maximum and minimum values of $|Z_1 - Z_2|$ is

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