Complex NumbersHard

Question

For all complex numbers z1, z2 satisfying |z1|=12 and |z2 - 3 - 4i | = 5, the minimum value of |z1 - z2 | is

Options

A.0
B.2
C.7
D.17

Solution

       
We know, |z1 - z2| = |z1 - (z2 - 3 - 4i) - (3 + 4i)|
        ≥ |z1| - |z2 - 3 - 4i| - |3 + 4i|
        ≥ 12 - 5 - 5 (using|z1 - z2 |≥|We know, |z1 - z2| = |z1 - (z2 - 3 - 4i) - (3 + 4i)|
        ≥ |z1| - |z2 - 3 - 4i| - |3 + 4i|
        ≥ 12 - 5 - 5 (using|z1 - z2 |≥| - |z2|)
∴      |z1 - z2| ≥ 2
Alternate Solution
Clearyl from the figure |z1 - z2| is minimum when z1, z2 lie along the diameter.
∴      |z1 - z2| ≥ C2B - C1A
               ∴ 12 - 10 = 2

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