Question

For all complex numbers z1, z2 satisfying l z1 l = 12 and l z2 - 3 - 4i l = 5, the minimum value of l z1 - z2 l is

Answer 1

As by triangle inequality
|z1-z2|≥|z1|-|z2|
and
|z2-3-4i|≥|z2|-|3+4i|
|z2-3-4i|≥|z2|-5
10≥|z2|
for minimum value
|z2|=10
|z1-z2|=|z1|-|z2|=12-10=2

Answer 2

Answer:

The minimum value of |Z₁-Z₂| is 2.

Explanation:

From the question we have,

|Z₁|=12                  (1)

|Z₂-3-4i|=5           (2)

To answer this question, we'll utilize the equation below:

$|Z_1-Z_2|\geq|Z_1| -|Z_2|$            (3)

We derive the following result by combining equations (1) and (2) in equation (3):

$|Z_2-3-4i| \geq |Z_2| -|3+4i|$

$|Z_2-3-4i| \geq |Z_2| -\sqrt{3^2+4^2}$

$|Z_2|\geq 10$       (4)

The smallest value of |Z₁-Z₂| ≥|Z₁| -|Z₂| is,

$|Z_1-Z_2|\geq 12-10$

$|Z_1-Z_2|\geq 2$

Hence, the minimum value of |Z₁-Z₂| is 2.

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