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If m and M denotes the minimum and maximum value of 2 z + 1 , where z - 2 i ≤ 1 and i 2 = - 1 , then the value of M - m 2 is equal to

Options

  1. A17
  2. B34
  3. C51
  4. D16

Correct answer

D. 16

Step-by-step solution

z - 2 i = 1 represents a circle with centre at 0 , 2 and radius 1 unit 2 z + 1 = 2 z - - 1 2 represents twice the distance from the point - 1 2 , 0 Hence from the diagram, m = 2 A D = 2 C D - A C m = 2 17 2 - 1 = 17 - 2 M = 2 B D = 2 C D + B C M = 2 17 2 + 1 = 17 + 2 ⇒ M - m 2 = 4 2 = 16

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