Let S = z ∈ C : z - 2 ≤ 1 , z 1 + i + z ¯ 1 - i ≤ 2 . Let z - 4 i attains minimum and maximum values, respectively, at z 1 ∈ S and z 2 ∈ S . If 5 z 1 2 + z 2 2 = α + β 5 , where α and β

Question

Let S = z &#8712; C : z - 2 &#8804; 1 , z 1 + i + z &#175; 1 - i &#8804; 2 . Let z - 4 i attains minimum and maximum values, respectively, at z 1 &#8712; S and z 2 &#8712; S . If 5 z 1 2 + z 2 2 = &#945; + &#946; 5 , where &#945; and &#946; are integers, then the value of &#945; + &#946; is equal to ______.

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Correct answer: 0

Given z - 2 &#8804; 1 x - 2 2 + y 2 &#8804; 1 &#160; &#160; &#160; &#160; &#160; &#8943; i represents interior region of a circle with centre 2 , 0 and radius 1 Now z 1 + i + z &#175; 1 - i &#8804; 2 Puting z = x + i y , we get x + i y 1 + i + x - i y 1 - i &#8804; 2 &#8658; x - y + i x + y + x - y - i x + y &#8804; 2 &#8756; &#160; x - y &#8804; 1 &#160; &#160; &#160; &#160; &#8230; i Let A 1 , 0 and B 2 , 1 be the points on the line x - y = 1 and the circle x - 2 2 + y 2 = 1 z - 4 i represents the distance from P

Let S = z ∈ C : z - 2 ≤ 1 , z 1 + i + z ¯ 1 - i ≤ 2 . Let z - 4 i attains minimum and maximum values, respectively, at z 1 ∈ S and z 2 ∈ S . If 5 z 1 2 + z 2 2 = α + β 5 , where α and β are integers, then the value of α + β is equal to ______.

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