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Question
For all complex numbers z1, z2 satisfying |z1|=20 and |z2−5−12i|=4, the minimum value of |z1−z2| is
For all complex numbers z1, z2 satisfying |z1|=20 and |z2−5−12i|=4, the minimum value of |z1−z2| is
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Solution
The correct option is B 3
|z1−z2|=|z1−(z2−5−12i)−(5+12i)|
|z1−z2|≥|z1|−|z2−5−12i|−|5+12i|
|z1−z2|≥20−4−13
|z1−z2|≥3
Alternative
From figure
∣∣z1−z2∣∣min=PQ
PQ=OP−OC−CQ⇒PQ=20−√52+122−4
∴PQ=∣∣z1−z2∣∣min=3
|z1−z2|=|z1−(z2−5−12i)−(5+12i)|
|z1−z2|≥|z1|−|z2−5−12i|−|5+12i|
|z1−z2|≥20−4−13
|z1−z2|≥3
Alternative
From figure ∣∣z1−z2∣∣min=PQ
PQ=OP−OC−CQ⇒PQ=20−√52+122−4
∴PQ=∣∣z1−z2∣∣min=3
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