Let S = z C : |z − 2| 1, z(1 + i) + z̄(1 − i) 2 . Let |z − 4i| attains minimum and maximum values, respectively, at z_ 1 S and z_ 2 S. If 5(|z_ 1 |^ 2 + |z_ 2 |^ 2 ) = + 5 , where and are integers, then the value of + is equal to _____.

JEE Main 2022 Mathematics question

Let S = {z C : |z − 2| , z(1 + i) + z̄(1 − 2}. Let |z − 4i| attains minimum and maximum values, respectively, at S and S. If , where and are integers, then the value of is equal to .

Answer: 26

Solution

The condition z(1 + i) + z̄(1 − is , i.e. . So S is the part of the disc on or above this line, a region bounded by the chord from B(1, 0) to C(2, 1) and an arc. The point of S nearest to 4i lies on the arc along the line joining 4i and the centre

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Question ID VK-035336