Question

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

• A. 75
• B. 30
• C. 35
• D. 20

Hint:

For problems involving max/min distances between points on circles ($|z-c_1|=r_1, |z-c_2|=r_2$), the key values are the distance between centers $d=|c_1-c_2|$, and the radii $r_1, r_2$. The max distance is $d+r_1+r_2$ and min distance is $|d-r_1-r_2|$ if they are exterior. For other configurations, a careful geometric or algebraic approach is needed.

Answer 1

The Correct Option is B

Step 1: Interpret the given equations geometrically.
The equation $|Z_1 - (3+4i)| = 5$ represents the set of all points $Z_1$ that lie on a circle with center $C_1 = 3+4i$ and radius $r_1 = 5$.
Similarly, the equation $|Z_2| = 15$ (or $|Z_2 - 0| = 15$) represents a circle centered at the origin $C_2 = 0$ with radius $r_2 = 15$.
The expression $|Z_1 - Z_2|$ denotes the distance between a point $Z_1$ on the first circle and a point $Z_2$ on the second circle.

Step 2: Use the triangle inequality to find the range of $|Z_1|$.
From $|Z_1 - (3+4i)| = 5$, we use the inequality $||a| - |b|| \le |a - b|$.
\[ ||Z_1| - |3+4i|| \le |Z_1 - (3+4i)|. \] Since $|3+4i| = \sqrt{3^2 + 4^2} = 5$, we get: \[ ||Z_1| - 5| \le 5. \] This gives: \[ -5 \le |Z_1| - 5 \le 5 \implies 0 \le |Z_1| \le 10. \] Hence, the possible range of $|Z_1|$ is $[0, 10]$.

Step 3: Use the triangle inequality again to find the range of $|Z_1 - Z_2|$.
We know that: \[ |Z_1 - Z_2| \le |Z_1| + |Z_2|. \] The maximum value occurs when both points are farthest apart: \[ \text{Max}(|Z_1 - Z_2|) = \text{max}(|Z_1|) + |Z_2| = 10 + 15 = 25. \] For the minimum distance: \[ |Z_1 - Z_2| \ge ||Z_1| - |Z_2||. \] However, to find the exact range, we use a geometric approach.
The distance between the centers of the circles is: \[ d = |C_1 - C_2| = |(3+4i) - 0| = 5. \] Let $U = Z_1 - (3+4i)$, so $|U| = 5$, and hence $Z_1 = U + (3+4i)$.
Then, \[ |Z_1 - Z_2| = |U + (3+4i) - Z_2| = |(3+4i) - (Z_2 - U)|. \] Let $V = Z_2 - U$. The range of $|V|$ is: \[ [|Z_2| - |U|, |Z_2| + |U|] = [15 - 5, 15 + 5] = [10, 20]. \] Now, we must find the range of $|(3+4i) - V|$ where $10 \le |V| \le 20$.
The minimum possible value is: \[ \text{Min}(|Z_1 - Z_2|) = |\,|V|_{\min} - |3+4i|\,| = 10 - 5 = 5. \] The maximum possible value is: \[ \text{Max}(|Z_1 - Z_2|) = |V|_{\max} + |3+4i| = 20 + 5 = 25. \]
Step 4: Calculate the sum of the maximum and minimum values.
\[ \text{Sum} = \text{Maximum value} + \text{Minimum value} = 25 + 5 = 30. \]