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Question
Let z1 and z2 be two complex numbers such that |z1|=12 and |z2−3−4i|=5. Then the minimum value of |z1−z2| is
Let z1 and z2 be two complex numbers such that |z1|=12 and |z2−3−4i|=5. Then the minimum value of |z1−z2| is
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Solution
The correct option is B 2
Given, |z1|=12 and |z2−3−4i|=5
|z1−z2|=|z1−(z2−3−4i)−(3+4i)|
Using the inequality,
|z1−z2|≥∣∣|z1|−|z2|∣∣ for any two complex numbers z1 and z2, we get
|z1−(z2−3−4i)−(3+4i)|≥||z1|−|z2−3−4i|−|3+4i||
⇒|z1−z2|≥|12−5−5|
⇒|z1−z2|≥2
Hence, the minimum value of |z1−z2| is 2.
Given, |z1|=12 and |z2−3−4i|=5
|z1−z2|=|z1−(z2−3−4i)−(3+4i)|
Using the inequality,
|z1−z2|≥∣∣|z1|−|z2|∣∣ for any two complex numbers z1 and z2, we get
|z1−(z2−3−4i)−(3+4i)|≥||z1|−|z2−3−4i|−|3+4i||
⇒|z1−z2|≥|12−5−5|
⇒|z1−z2|≥2
Hence, the minimum value of |z1−z2| is 2.
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