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Question
If z is any complex number satisfying |z−3−2i|≤2, then the minimum value of |2z−6+5i| is
If z is any complex number satisfying |z−3−2i|≤2, then the minimum value of |2z−6+5i| is
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Solution
The correct option is A 5
Given |z−3−2i|≤2 ⋯(1)
Now, |2z−6+5i|
=|2(z−3−2i)+9i|
≥||2(z−3−2i)|−|9i|| (∵|z1+z2|≥||z1|−|z2||)
⇒|2z−6+5i|≥|2|z−3−2i|−9|
From equation (1),
|2z−6+5i|≥5
Given |z−3−2i|≤2 ⋯(1)
Now, |2z−6+5i|
=|2(z−3−2i)+9i|
≥||2(z−3−2i)|−|9i|| (∵|z1+z2|≥||z1|−|z2||)
⇒|2z−6+5i|≥|2|z−3−2i|−9|
From equation (1),
|2z−6+5i|≥5
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