Question

For any complex number z, the minimum value of |z|+|z-2i| is

Answer 1

Step-by-step explanation:

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Answer 2

Therefore the minimum value of $|z|+|z-2i|$ is $2$

Step-by-step explanation:

Given;

Find the minimum value of $|z|+|z-2i|$

For finding minimum value of a modulus complex number we know;

$\left | a-b \right |\leq \left | a\right |+\left | b\right |\leq \left | a+b\right |$

By using this formula,

$\left | z-z+2i \right |\leq \left | z\right |+\left | z-2i\right |\leq \left | z+z-2i\right |$

$\left |2i \right |\leq \left | z\right |+\left | z-2i\right |\leq \left | 2z-2i\right |$

From above equation,

$2\leq \left | z\right |+\left | z-2i\right |$  (∵ $\left | 2i \right |=\sqrt{4} =2$ )

So the minimum value of $|z|+|z-2i|$ is $2$