complex inequality

[devout stratus]

How to solve this one with short method?

[viral lily]

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[fickle seal]

Focus on the fact that |z| is just the distance between point z and origin.
Thus, |z1| = |z1 - 0|.
Similarly for |z2|.

Finally, think of triangle inequality.

[devout stratus]

It seems C

[fickle seal]

No. That doesn't seem right.

[haughty socket]

you just calculate

$$ \sqrt{a^2+b^2} = \sqrt{c^2+d^2} + \sqrt{(a-c)^2 + (b-d)^2} $$

[fierce bramble]

aL

[haughty socket]

and now conclude something

or you could note that

$$ |z| -|w|\leqslant |z-w| $$

[fierce bramble]

aL

[haughty socket]

this holds always due to triangle inequality

so you just have to worry about the converse inequality

@devout stratus

and if it's none of these, think of counterexamples to each option that is not d)

[devout stratus]

Option a is correct

[devout stratus]

@haughty socket

I want to understand this inequality more