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已知 z1z_{1} 为复数，且 ∣z1∣=2\left|z_{1}\right|=2 ，则 ∣z1+2i∣\left|z_{1}+2 \mathrm{i}\right| 的最大...
Mar 8, 2024
已知 z1z_{1} 为复数，且 ∣z1∣=2\left|z_{1}\right|=2 ，则 ∣z1+2i∣\left|z_{1}+2 \mathrm{i}\right| 的最大值为
Solution by Steps
step 1
To find the maximum value of ∣z1+2i∣ |z_1 + 2i| given ∣z1∣=2 |z_1| = 2 , we use the triangle inequality
step 2
The triangle inequality states that ∣z1+z2∣≤∣z1∣+∣z2∣ |z_1 + z_2| \leq |z_1| + |z_2|
step 3
Applying the inequality with z1 z_1 and 2i 2i , we get ∣z1+2i∣≤∣z1∣+∣2i∣ |z_1 + 2i| \leq |z_1| + |2i|
step 4
Since ∣z1∣=2 |z_1| = 2 and ∣2i∣=2 |2i| = 2 , the inequality becomes ∣z1+2i∣≤2+2 |z_1 + 2i| \leq 2 + 2
step 5
Therefore, the maximum value of ∣z1+2i∣ |z_1 + 2i| is 4 when z1 z_1 is in the direction opposite to 2i 2i , which is z1=−2 z_1 = -2
Answer
4
Key Concept
Triangle Inequality
Explanation
The triangle inequality is used to find the maximum value of the modulus of a complex number sum.
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