3
Question
For all complex numbers z1,z2 satisfying |z1|=12 and |z2−3−4i|=5, the minimum value of |z1−z2| is
For all complex numbers z1,z2 satisfying |z1|=12 and |z2−3−4i|=5, the minimum value of |z1−z2| is
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Solution
The correct option is A 2
|z1−z2|=|z1−(z2−3−4i)−(3+4i)|≥|z1|−|z2−3−4i|−|3+4i|≥12−5−5 [using |z1−z2|≥|z1|−|z2|]
Alternate Solution
Clearly from the figure |z1−z2| is minimum when z1,z2 lie along the diameter.
∵|z1−z2|≥C2B−C2A⟹|z1−z2|≥12−10=2
|z1−z2|=|z1−(z2−3−4i)−(3+4i)|≥|z1|−|z2−3−4i|−|3+4i|≥12−5−5 [using |z1−z2|≥|z1|−|z2|]
Alternate Solution
Clearly from the figure |z1−z2| is minimum when z1,z2 lie along the diameter.
∵|z1−z2|≥C2B−C2A⟹|z1−z2|≥12−10=2
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